CTW: Mathematical and Computational Challenges in Cilia- and Flagella-Induced Fluid Dynamics: Titles & Abstracts
Slender-body theories allow for the representation of thin tubes in Stokes' flow by a distribution of fundamental solutions along the filament center line while approximately enforcing boundary conditions on the surface of the tube. The idea is revisited here in the more general setting of regularized forces in a small neighborhood of the center line. The regularity in the forces produces a smooth final expression that helps eliminate the computational instabilities of the unregularized formulas. The derivations of the regular slender body theories corresponding with the standard theories of Lighthill and of Keller and Rubinow are outlined. Consistency with these theories is verified in the limit as the smoothing parameter vanishes. Numerical issues of the resulting theories are addressed in the context of test problems.
This work has been a collaboration with Michael Nicholas of the Colorado School of Mines.
The motility of sperm flagella and cilia are based on a common axonemal structure. This structure is capable of generating a wide range of dynamical behavior modulated by signaling molecules as well as the properties of the fluid environment. We describe a fluid-mechanical model for the axoneme coupling the internal force generation of dynein molecular motors through the passive elastic axonemal structure with the external fluid mechanics. As shown in numerical simulations, the model's flagellar waveform depends strongly on viscosity as well as dynein strength. We show an extension of our original model for Newtonian fluids to complex viscoelastic fluids in order to model mucus transport by cilia in the respiratory tract as well as sperm motility in reproduction. These immersed boundary models for sperm and ciliary motility in complex fluids explore continuum approaches such as Oldroyd-B as well as Lagrangian moving mesh methods.
The emergence of single molecule experimental techniques coupled with the development of in vitro motility assays has revolutionized our knowledge of how isolated molecular motors convert chemical energy from ATP hydrolysis into a continuous linear movement along microtubules or actin filaments. However, biology abounds with examples ranging from periodic beating of eukaryotic cilia to macroscopic contraction of skeletal muscle wherein thousands of molecular motors coordinate their movement on molecular lengthscales to produce entirely new dynamics at macroscopic scales. Studying such emergent phenomena presents significant experimental challenges but also an opportunity to gain insight into fundamental biological processes while simultaneously uncovering fundamental physics of systems that are driven to highly out-of-equilibrium states. In this vein, our group has focused on reconstituting far-from-equilibrium structures from purified biochemical components. I will describe recent advances in this area including: (1) assembly of a minimal model of synthetic cilia capable of generating periodic beating patterns, (2) study of 2D active liquid crystals and (3) reconstitution of cytoplasmic streaming within micron sized droplets.
The beating of a cilium is an elegant example of an actuated elastic structure coupled to a surrounding fluid. Computational fluid dynamics enthusiasts will recognize that ciliary systems present many complications such as the interaction of groups of cilia, the influence of boundaries, and the coupling to fluids that have complex rheology and microstructures. Moreover, the ciliary beatform is an emergent feature of these mechanical considerations along with biochemical processes. We will present an overview of current CFD models of cilia, along with some recent progress in analyzing fluid mixing by cilia and modeling ciliary penetration of a mucus layer.
Since the pioneering studies of GI Taylor in the fifties, models have been used to gain understanding in the propulsion of microorganisms. Modern microfabrication techniques enable us to assemble very small scale devices emulating the motion of cilia. I will review the different strategies used in recent years towards the goal of fabricating micron scale artificial swimmers. In particular I will discuss the relative merits of self-assembly and micromolding. I will describle several sources of propulsive energy but most of the talk will be devoted to magnetically driven systems.
Maneuvering nanoscale objects in fluidic media in a non-invasive manner can lead to various biomedical applications, and is pursued by researchers across many disciplines. Of particular interest is the possibility of powering and controlling the motion of nanoscale objects with small, homogeneous magnetic fields, which is easy to achieve, and guaranteed to be non-invasive as well. This has recently been achieved by various groups using advanced nanofabrication techniques, where magnetic nanoscale objects of different shapes, such as helical, flexible rod-likeetc. have been maneuvered in a controllable fashion using either rotating or undulating magnetic fields. In particular, cork-screw motion is achieved in ferromagnetic helical nanostructures by aligning the permanent magnetic moments of the helix with a rotating magnetic field, causing the nanostructure to rotate and therefore propel. Such systems have been referred to as either magnetic nanopropellers or as artificial bacterial flagella in the literature. In this talk, we will discuss the fabrication and actuation of such a system, and describe their complex dynamical behavior in the presence of thermal fluctuations. In particular, we will describe how this novel system can show bistable dynamics and may have non-gaussian speed fluctuations under certain conditions.
Microorganisms and the mechanical components of the cell motility machinery such as cilia and flagella operate in low Reynolds number conditions where hydrodynamics is dominated by viscous forces. The medium thus induces a long-ranged hydrodynamic interaction between these active objects, which could lead to synchronization, coordination and other emergent many-body behaviors. In my talk, I will examine these effects using minimal models that are simple enough to allow extensive analysis that sheds light on the underlying mechanisms for the emergent phenomena.
The planarian ventral surface is a completely exposed ciliated epithelium. These animals utilize their motile cilia to generate gliding locomotion by beating against secreted mucus. The ventral cilia have a standard 9+2 axoneme containing both inner and outer rows of dynein arms and beat at ~22 Hz. RNAi knockdown approaches are simple and robust, leading to reductions in mRNA to almost undetectable levels. Thus, planaria may be used to rapidly screen proteins of unknown function for their role in ciliary assembly and/or motility, and may also provide a useful model system in which to investigate muco-ciliary interactions. I will discuss the use of this organism to analyze the role of outer arm dynein components in generating motile force and in maintaining the hydrodynamic coupling required for metachronal synchrony of beating cilia.
The ventricular system in the brain is lined by multiciliated cells. The motility of these ependymal cilia was analyzed in hy3-/- mice which carry a null mutation in Hydin and develop lethal hydrocephalus. Hy3-/- cilia lack a projection from the ciliary central pair and move with slightly reduced beat frequency and a greatly reduced beat amplitude. They lack the ability to generate fluid flow explaining the hydrocephalic phenotype of the mutant mice. The assembly of motile and non-motile cilia requires intraflagellar transport (IFT) but it remains largely unknown how IFT traffics ciliary precursors. Simultaneous in vivo imaging of IFT and cargoes revealed a complex pattern of IFT and non-IFT cargo movements, and unloading and assembly site docking events. Quantitative data on cargo frequency, assembly, and turn-over will provide a basis for future modeling of ciliary assembly and dynamics.
In this talk, I will introduce a regularization method that gives a smooth formulation for the fundamental solution to Stokes flow driven by an infinite, triply-periodic array of point forces. With this formulation, the velocity at any spatial location may be calculated, including at and very near the point forces; these locations typically lead to numerical difficulties due to the singularity within the Stokeslet when using other methods. For computational efficiency, the current method is built upon previous methods in which the periodic Stokeslet is split into two rapidly decaying sums, one in physical space and one in reciprocal, or Fourier, space. I will show a few validation studies and then discuss a recent extension of the method to doubly-periodic flow. Finally, using the extended method, simulations of doubly-periodic arrays of beating cilia will be presented.
Flagellar bundling is an important aspect of locomotion in bacteria such as Escherichia coli. To study the hydrodynamic behavior of helical flagella, we present a computational model that is based on a generalized version of the immersed boundary method combined with the nonstandard Kirchhoff rod theory. We consider two model flagella, each of which has a rotary motor at its base with the rotation rate of the motor set at 100 Hz. Bundling occurs when both flagella are left-handed helices turning counterclockwise (when viewed from the nonmotor end of the flagellum looking back toward the motor) or when both flagella are right-handed helices turning clockwise. Helical flagella of the other combinations of handedness and rotation direction do not bundle.
A fluid-structure interaction model that couples viscoelastic fluid motion induced by collective behavior of cilia to detailed axoneme mechanics is used to investigate bounds on cilia structure and mucus properties that determine effective fluid clearance. Dynein forcing is represented by a stochastic walker model that responds to local ATP concentrations provided by a biochemical network model. Cilium axonemes are modeled by large-deflection finite elements representing microtubules and viscoelastic springs representing connecting elements (nexins, radial spokes). Cilium motion is coupled to a viscoelastic fluid computation that models gel-like behavior of the mucus as well as possible Newtonian behavior of the periciliary fluid layer. Behavior of the viscoelastic fluid is prescribed at a microscopic level to avoid using continuum viscoelastic models of questionable validity. A lattice-based technique based upon a variational formulation of the Fokker-Planck equation is used to describe the viscoelastic fluid dynamics. A lattice Boltzmann method is applied to capture forcing of the viscoelastic mucus layer by concurrent airflow. The overall model exhibits natural formation of metachronal waves due to phase coupling of the dynein motion. Adjoint density analysis and uncertainty quantification techniques are applied to assess the stability of the transport induced by metachronal waves to perturbations in dynein walker rates, axoneme element rigidity, and mucus gel-formation process. The goal is not only to assess the robustness of the metachronal transport process, but also to identify elements within the overall transport mechanism that are most promising targets for pharmaceutical treatment of ciliary dysfunction.
Sperm are known to exhibit two distinct types of motility. One is characterized by constant amplitude, symmetrical waveforms. The other is characterized by asymmetrical waveforms, which are correlated with an increase in calcium concentration. The goal of this work is to model the undulatory swimming of sperm swimming in a viscous, incompressible fluid using the method of regularized Stokeslets. Varying waveforms will be considered via a preferred curvature function. Results showing emergent waveforms, swimming speeds, and trajectories will be compared to experimental data.
Primary cilia are able to sense flow. This is these years a banality ñ or is it? The question has been addressed with quite many approaches. And at least for a long row of mechano-sensitive cells, especially epithelial cells with long cilia this seem to hold. The primary cilium is not in itself necessary for a cell to be mechano-sensitive. If the stimulus were substantial enough any cell would react to mechanical stimulation. The cilium merely sensitise a given cell to mechanical stimuli ñ and that gives a challenge when one wants to address the ciliary effects. First of all they are subtle, which means that one has to titrate the system to catch these effects without mechanically over-stimulating the cells. Nevertheless, when addressing the sensory functions of cilia, mechanical or receptory ñ less is more. The talk is a short overview over the cilum as a flow censor and how to catch the signal.
Many microorganisms swim by rotating one or many helical flagella, often propelling themselves through fluids that exhibit both viscous and elastic qualities in response to deformations. In an effort to better understand the complex interaction between the fluid and body in such systems, we have studied numerically the force-free swimming of a rotating helix in a viscoelastic (Oldroyd-B) fluid. The introduction of viscoelasticity can either enhance or retard the swimming speed depending on the body geometry and the properties of the fluid (through a dimensionless Deborah number). The results are compared to recent experiments on a rotating helix immersed in a Boger fluid. Our findings bridge the gap between studies showing situationally dependent enhancement or retardation of swimming speed, and may help to clarify phenomena observed in a number of biological systems.
In microfluidic applications, the Reynolds number is often very small, and the dynamics of the fluid can be described by the Stokes equations, which can be reformulated as a boundary integral equation.
Numerical simulations based on boundary integral formulations can be accelerated using a fast summation method. I will present a spectrally accurate FFT based Ewald method for this purpose. This method allows for the use of much smaller FFT grids, as compared to established methods. The method has been adopted to the simulation of rigid fiber suspensions, and modified to allow for analytic integration for fibers that are close. Due to the relative smallness of the FFT grids, it is possible to treat larger periodic domains, including a larger number of fibers, and still fit in on a desktop. I will also discuss the extension of our spectral Ewald method to the case of planar periodicity (periodic in two of the three dimensions), a case for which no fast Ewald methods previously existed for Stokes.
Posters
We propose that the rheological properties of background fluid play an important role in the interaction of microorganisms with the flow field. The viscoelastic-induced migration of microorganisms in a vortical flow leads to the emergence of a limit cycle. The shape and formation rate of patterns depend on motility, vorticity strength, and rheological properties of the background fluid. Given the inherent viscoelasticity of exopolysaccharides secreted by microorganisms, our results can suggest new mechanisms leading to the vital behavior of microorganisms such as bacterial aggregation and biofilm formation.
Small planktonic organisms ubiquitously display unsteady or impulsive motion to attack a prey or escape a predator in natural environments. Despite this, the role of unsteady forces such as history and added mass forces on the low Reynolds number propulsion of small organisms, e.g. Paramecium, is poorly understood. In this work, we derive the fundamental equation of motion for an organism swimming by the means of surface distortion in a non-uniform background flow field at a low Reynolds number regime. We show that the history and added mass forces are important as the Stokes number increases above unity.
Starting from coarse grained models of tubulin dimers, various finite element models are formulated and compared to determine accuracy of deformation-force behavior. Stiffness values are compared to processed experimental data. Conclusions for overall cilium axoneme models are presented.
We develop a 2D numerical model to simulate muco-cilia interactions. The cilia are modeled as elastic beams under the plane strain assumption. The motion of cilia is solved using the Material Point Method developed by Sulsky(1994). The mucus layer is assumed to be Newtonian, governed by the incompressible Navier-Stokes equations. We present some preliminary simulations.
Motile cilia play a large role in fluid motion across the surface of ciliated tissue. We present an experimental and theoretical study involving a single rigid cilium rotating in a viscous fluid about one of its ends in contact with a horizontal no-slip plane. Experimentally tracked three dimensional Lagrangian trajectories are compared with theoretical trajectories computed using a properly imaged slender body theory. The addition of planar bend to the rod geometry is shown to break symmetry and create large scale nested tori in the Lagrangian particle trajectories. Three dimensional PIV measurements are presented which help to explain the origin of the large scale tori and compared directly with the slender body theory.
