Sookkyung Lim

Simulations of the Whirling Instability by the Immersed Boundary Method

When an elastic filament spins in a viscous incompressible fluid it may undergo a whirling instability, as studied asymptotically by Wolgemuth, Powers, and Goldstein [Phys. Rev. Lett., 84 (2000), pp. 16--23]. We use the immersed boundary (IB) method to study the interaction between the elastic filament and the surrounding viscous fluid as governed by the incompressible Navier--Stokes equations. This allows the study of the whirling motion when the shape of the filament is very different from the unperturbed straight state.

Dynamics of a Closed Rod with Twist and Bend in Fluid

We investigate the instability and subsequent dynamics of a closed rod with twist and bend in a viscous, incompressible fluid. A new version of the immersed boundary (IB) method is used in which the immersed boundary applies torque as well as force to the surrounding fluid and in which the equations of motion of the immersed boundary involve the local angular velocity as well as the local linear velocity of the fluid. An important feature of the IB method in this context is that self-crossing of the rod is automatically avoided because the rod moves in a continuous (interpolated) velocity field. A rod with a uniformly distributed twist that has been slightly perturbed away from its circular equilibrium configuration is used as an initial condition, with the fluid initially at rest. If the twist in the rod is sufficiently small, the rod simply returns to its circular equilibrium configuration, but for larger twists that equilibrium configuration becomes unstable, and the rod undergoes large excursions before relaxing to a stable coiled configuration.

Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation

We develop a Lagrangian numerical algorithm for an elastic rod immersed in a viscous, incompressible fluid at zero Reynolds number. The elasticity of the rod is described by a version of the Kirchhoff rod model, where intrinsic curvature and twist are prescribed, and the fluid is governed by the Stokes equations in R^3. The elastic rod is represented by a space curve corresponding to the centerline of the rod and an orthonormal triad, which encodes the bend and twist of the rod. In this method, the differences between the rod configuration and its intrinsic shape generate force and torque along the centerline. The coupling to the fluid is accomplished by the use of the method of regularized Stokeslets for the force and regularized rotlets for the torque. This technique smooths out the singularity in the fundamental solutions of the Stokes equations for the computation of the velocity of the rod centerline. In addition, the computation of the angular velocity of the rod requires the use of regularized (potential) dipoles. As a benchmark problem, we consider open and closed rods with intrinsic curvature and twist in a viscous fluid. Equilibrium configurations and dynamic instabilities are compared with known results in elastic rod theory. For cases when the exact solution is unknown, the numerical results are compared to those produced by the generalized immersed boundary (gIB) method, where the fluid is governed by the Navier-Stokes equations with small Reynolds number on a finite (periodic) domain. It is shown that the regularization method combined with Kirchhoff rod theory contributes substantially to the reduction of computation time and efficient memory usage in comparison to the gIB method. We also illustrate how the regularized method can be used to model microorganism motility where the organism is propelled by a flagellum propagating sinusoidal waves. The swimming speeds of this flagellum using the regularized Stokes formulation are matched well with classical asymptotic results of Taylors infinite cylinder in terms of frequency and amplitude of the undulation.

Blood Flow in a Compliant Vessel by the Immersed Boundary Method

In this paper we develop a computational approach to analyze hemodynamics in the aorta; this may serve as a useful tool in the development of noninvasive methods to detect early onset of diseases such as aneurysms and stenosis in major blood vessels. We introduce a mathematical model which describes the interaction of blood flow with the aortic wall; this model is based on the immersed boundary method. A two-dimensional vessel model is constructed, the velocity at the inlet is prescribed based on the information from the Magnetic Resonance Imaging data measured in the aorta of a healthy subject, and the velocity at the outlet is prescribed by driving the pressure level reproduced from the literature. The mathematical model is validated by comparing with well-known solutions of the viscous incompressible Navier–Stokes equations, i.e., Womersley flow. The hysteresis behavior in the pressure–diameter relation is observed when the viscoelastic material property of the arterial wall is taken into consideration. Five different shapes of aortic wall are considered for comparison of the flow patterns inside the aorta: one for the normal aorta, two for the dilated aorta, and two for the constrictive aorta.

Dynamics of an open elastic rod with intrinsic curvature and twist in a viscous fluid

A twisted elastic rod with intrinsic curvature is considered. We investigate the dynamics of the rod in a viscous incompressible fluid. This fluid is governed by the Navier–Stokes equations and the fluid-rod interaction problem is solved by the generalized immersed boundary method combined with the Kirchhoff rod theory. We classify the equilibrium configurations of an open elastic rod as they depend on the rod’s intrinsic characteristics and fluid properties. We assume that the intrinsic curvature and twist are distributed uniformly along the rod. In the case of zero intrinsic curvature (i.e., the stress-free state of the rod is straight), we find a critical value of twist, below which the straight state of the rod is stable. When the twist is above this critical value, however, the rod buckles locally and produces a loop or a plectoneme or a combination of both. When the constant intrinsic curvature is nonzero, we also find a critical value of twist that distinguishes a buckled rod from a stable helix. W...

Robustness and period sensitivity analysis of minimal models for biochemical oscillators.

Biological systems exhibit numerous oscillatory behaviors from calcium oscillations to circadian rhythms that recur daily. These autonomous oscillators contain complex feedbacks with nonlinear dynamics that enable spontaneous oscillations. The detailed nonlinear dynamics of such systems remains largely unknown. In this paper, we investigate robustness and dynamical differences of five minimal systems that may underlie fundamental molecular processes in biological oscillatory systems. Bifurcation analyses of these five models demonstrate an increase of oscillatory domains with a positive feedback mechanism that incorporates a reversible reaction, and dramatic changes in dynamics with small modifications in the wiring. Furthermore, our parameter sensitivity analysis and stochastic simulations reveal different rankings of hierarchy of period robustness that are determined by the number of sensitive parameters or network topology. In addition, systems with autocatalytic positive feedback loop are shown to be more robust than those with positive feedback via inhibitory degradation regardless of noise type. We demonstrate that robustness has to be comprehensively assessed with both parameter sensitivity analysis and stochastic simulations.

Three-Dimensional Simulations of a Closed Valveless Pump System Immersed in a Viscous Fluid

We present a three-dimensional model of flow driven by pumping without valves (valveless pumping) in a closed loop system in which the closed loop of tubing is immersed in an incompressible viscous fluid. This closed tube consists of two parts, an open cylindrical soft tube and an open rigid tube, smoothly connected to one another. At an asymmetric location of the soft tube, a periodic compress-and-release action with a time delay between actions is taken to create a net flow. Numerical results show that the magnitude of average net flow and flow direction inside the tube depend on the pumping frequency, the amplitude of periodic forcing, the compression duration, the length of the soft tube, and the elastic properties of the tube. Fluid viscosity is also found to influence net flow. The immersed boundary method is used to investigate the interaction between the tube and the fluid and to study the valveless pumping mechanism. (A correction to the this article has been appended at the end of the pdf file.)

The role of myosin II in glioma invasion: A mathematical model

Gliomas are malignant tumors that are commonly observed in primary brain cancer. Glioma cells migrate through a dense network of normal cells in microenvironment and spread long distances within brain. In this paper we present a two-dimensional multiscale model in which a glioma cell is surrounded by normal cells and its migration is controlled by cell-mechanical components in the microenvironment via the regulation of myosin II in response to chemoattractants. Our simulation results show that the myosin II plays a key role in the deformation of the cell nucleus as the glioma cell passes through the narrow intercellular space smaller than its nuclear diameter. We also demonstrate that the coordination of biochemical and mechanical components within the cell enables a glioma cell to take the mode of amoeboid migration. This study sheds lights on the understanding of glioma infiltration through the narrow intercellular spaces and may provide a potential approach for the development of anti-invasion strategies via the injection of chemoattractants for localization.

Modeling polymorphic transformation of rotating bacterial flagella in a viscous fluid

The helical flagella that are attached to the cell body of bacteria such as Escherichia coli and Salmonella typhimurium allow the cell to swim in a fluid environment. These flagella are capable of polymorphic transformation in that they take on various helical shapes that differ in helical pitch, radius, and chirality. We present a mathematical model of a single flagellum described by Kirchhoff rod theory that is immersed in a fluid governed by Stokes equations. We perform numerical simulations to demonstrate two mechanisms by which polymorphic transformation can occur, as observed in experiments. First, we consider a flagellar filament attached to a rotary motor in which transformations are triggered by a reversal of the direction of motor rotation [L. Turner et al., J. Bacteriol. 182, 2793 (2000)10.1128/JB.182.10.2793-2801.2000]. We then consider a filament that is fixed on one end and immersed in an external fluid flow [H. Hotani, J. Mol. Biol. 156, 791 (1982)10.1016/0022-2836(82)90142-5]. The detailed dynamics of the helical flagellum interacting with a viscous fluid is discussed and comparisons with experimental and theoretical results are provided.

Dynamical simulations of DNA supercoiling and compression

In the present article, we summarize our recent studies of DNA dynamics using the generalized immersed boundary method. Our analysis of the effects of electrostatic repulsion on the dynamics of DNA supercoiling revealed that, after perturbation, a pre-twisted DNA collapses into a compact supercoiled configuration that is sensitive to the initial excess link and ionic strength of the solvent. A stochastic extension of the generalized immersed boundary method shows that DNA in solution subjected to a constant electric field is compressed into a configuration with smaller radius of gyration and smaller ellipticity ratio than those expected for such a molecule in a thermodynamic equilibrium.