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Colloquium: Biophysical principles of undulatory self-propulsion in granular media

(Georgia Institute of Technology, 2014) Goldman, Daniel I.

Biological locomotion, movement within environments through self-deformation, encompasses a range of time and length scales in an organism. These include the electrophysiology of the nervous system, the dynamics of muscle activation, the mechanics of the skeletal system, and the interaction mechanics of such structures within natural environments like water, air, sand, and mud. Unlike the many studies of cellular and molecular scale biophysical processes, movement of entire organisms (like flies, lizards, and snakes) is less explored. Further, while movement in fluids like air and water is also well studied, little is known in detail of the mechanics that organisms use to move on and within flowable terrestrial materials such as granular media, ensembles of small particles that collectively display solid, fluid, and gaslike behaviors. This Colloquium reviews recent progress to understand principles of biomechanics and granular physics responsible for locomotion of the sandfish, a small desert-dwelling lizard that “ swims” within sand using undulation of its body. Kinematic and muscle activity measurements of sand swimming using high speed x-ray imaging and electromyography are discussed. This locomotion problem poses an interesting challenge: namely, that equations that govern the interaction of the lizard with its environment do not yet exist. Therefore, complementary modeling approaches are also described: resistive force theory for granular media, multiparticle simulation modeling, and robotic physical modeling. The models reproduce biomechanical and neuromechanical aspects of sand swimming and give insight into how effective locomotion arises from the coupling of the body movement and flow of the granular medium. The argument is given that biophysical study of movement provides exciting opportunities to investigate emergent aspects of living systems that might not depend sensitively on biological details.
The Effectiveness of Resistive Force Theory in Granular Locomotion

(Georgia Institute of Technology, 2014) Zhang, Tingnan ; Goldman, Daniel I.

Resistive force theory (RFT) is often used to analyze the movement of microscopic organisms swimming in fluids. In RFT, a body is partitioned into infinitesimal segments, each of which generates thrust and experiences drag. Linear superposition of forces from elements over the body allows prediction of swimming velocities and efficiencies. We show that RFT quantitatively describes the movement of animals and robots that move on and within dry granular media (GM), collections of particles that display solid, fluid, and gas-like features. RFT works well when the GM is slightly polydisperse, and in the “frictional fluid” regime such that frictional forces dominate material inertial forces, and when locomotion can be approximated as confined to a plane. Within a given plane (horizontal or vertical) relationships that govern the force versus orientation of an elemental intruder are functionally independent of the granular medium. We use the RFT to explain features of locomotion on and within granular media including kinematic and muscle activation patterns during sand-swimming by a sandfish lizard and a shovel-nosed snake, optimal movement patterns of a Purcell 3-link sand-swimming robot revealed by a geometric mechanics approach, and legged locomotion of small robots on the surface of GM. We close by discussing situations to which granular RFT has not yet been applied (such as inclined granular surfaces), and the advances in the physics of granular media needed to apply RFT in such situations.
Effect of Volume Fraction on Granular Aavalanche Dynamics

(Georgia Institute of Technology, 2014) Gravish, Nick ; Goldman, Daniel I.

We study the evolution and failure of a granular slope as a function of prepared volume fraction, φ0. We rotated an initially horizontal layer of granular material (0.3-mm-diam glass spheres) to a 45◦ angle while we monitor the motion of grains from the side and top with high-speed video cameras. The dynamics of grain motion during the tilt process depended sensitively on φ0∈ [0.58–0.63] and differed above or below the granular critical state, φc, defined as the onset of dilation as a function of increasing volume fraction. For φ0−φc < 0, slopes experienced short, rapid, precursor compaction events prior to the onset of a sustained avalanche. Precursor compaction events began at an initial angle θ0 = 7.7 ± 1.4◦ and occurred intermittently prior to the onset of an avalanche. Avalanches occurred at the maximal slope angle θm =28.5 ± 1.0◦. Granular material at φ0 − φc > 0 did not experience precursor compaction prior to avalanche flow, and instead experienced a single dilational motion at θ0 = 32.1 ± 1.5◦ prior to the onset of an avalanche at θm = 35.9 ± 0.7◦. Both θ0 and θm increased with φ0 and approached the same value in the limit of random close packing. The angle at which avalanching grains came to rest, θR = 22 ± 2◦, was independent of φ0. From side-view high-speed video, we measured the velocity field of intermittent and avalanching flow. We found that flow direction, depth, and duration were affected by φ0, with φ0 − φc < 0 precursor flow extending deeper into the granular bed and occurring more rapidly than precursor flow at φ0 − φc > 0. Our study elucidates how initial conditions—including volume fraction—are important determinants of granular slope stability and the onset of avalanches.
Force and flow at the onset of drag in plowed granular media

(Georgia Institute of Technology, 2014) Gravish, Nick ; Umbanhowar, Paul B. ; Goldman, Daniel I.

We study the transient drag force F[subscript D] on a localized intruder in a granular medium composed of spherical glass particles. A flat plate is translated horizontally from rest through the granular medium to observe how F[subscript D] varies as a function of the medium’s initial volume fraction, φ. The force response of the granular material differs above and below the granular critical state, φ[subscript c], the volume fraction which corresponds to the onset of grain dilatancy. For φ<φ[subscript c] F[subscript D] increases monotonically with displacement and is independent of drag velocity for the range of velocities examined (<10 cm/s). For φ>φ[subscript c], F[subscript D] rapidly rises to a maximum and then decreases over further displacement. The maximum force for φ>φ[subscript c] increases with increasing drag velocity. In quasi-two-dimensional drag experiments, we use granular particle image velocimetry (PIV) to measure time resolved strain fields associated with the horizontal motion of a plate started from rest. PIV experiments show that the maxima in F[subscript D] for φ>φ[subscript c] are associated with maxima in the spatially averaged shear strain field. For φ>φ[subscript c] the shear strain occurs in a narrow region in front of the plate, a shear band. For φ<φ[subscript c] the shear strain is not localized, the shear band fluctuates in space and time, and the average shear increases monotonically with displacement. Laser speckle measurements made at the granular surface ahead of the plate reveal that for φ<φ[subscript c] particles are in motion far from the intruder and shearing region. For φ>φ[subscript c], surface particles move only during the formation of the shear band, coincident with the maxima in F[subscript D], after which the particles remain immobile until the sheared region reaches the measurement region.
High-resolution mapping of h1 linker histone variants in embryonic stem cells.

(Georgia Institute of Technology, 2013-04) Cao, Kaixiang ; Lailler, Nathalie ; Zhang, Yunzhe ; Kumar, Ashwath ; Uppal, Karan ; Liu, Zheng ; Lee, Eva K. ; Wu, Hongwei ; Medrzycki, Magdalena ; Pan, Chenyi ; Ho, Po-Yi ; Cooper, Guy P. , Jr. ; Dong, Xiao ; Bock, Christoph ; Bouhassira, Eric E. ; Fan, Yuhong

H1 linker histones facilitate higher-order chromatin folding and are essential for mammalian development. To achieve highresolution mapping of H1 variants H1d and H1c in embryonic stem cells (ESCs), we have established a knock-in system and shown that the N-terminally tagged H1 proteins are functionally interchangeable to their endogenous counterparts in vivo. H1d and H1c are depleted from GC- and gene-rich regions and active promoters, inversely correlated with H3K4me3, but positively correlated with H3K9me3 and associated with characteristic sequence features. Surprisingly, both H1d and H1c are significantly enriched at major satellites, which display increased nucleosome spacing compared with bulk chromatin. While also depleted at active promoters and enriched at major satellites, overexpressed H10 displays differential binding patterns in specific repetitive sequences compared with H1d and H1c. Depletion of H1c, H1d, and H1e causes pericentric chromocenter clustering and de-repression of major satellites. These results integrate the localization of an understudied type of chromatin proteins, namely the H1 variants, into the epigenome map of mouse ESCs, and we identify significant changes at pericentric heterochromatin upon depletion of this epigenetic mark.
On a transformation of Bohl and its discrete analogue

(Georgia Institute of Technology, 2013) Harrell, Evans M. ; Wong, Manwah Lilian

Fritz Gesztesy’s varied and prolific career has produced many transformational contributions to the spectral theory of one-dimensional Schrödinger equations. He has often done this by revisiting the insights of great mathematical analysts of the past, connecting them in new ways, and reinventing them in a thoroughly modern context. In this short note we recall and relate some classic transformations that figure among Fritz Gestesy’s favorite tools of spectral theory, and indeed thereby make connections among some of his favorite scholars of the past, Bohl, Darboux, and Green. After doing this in the context of one-dimensional Schrödinger equations on the line, we obtain some novel analogues for discrete one-dimensional Schrödinger equations.
Mechanics of undulatory swimming in a frictional fluid

(Georgia Institute of Technology, 2012-12) Ding, Yang ; Sharpe, Sarah S. ; Masse, Andrew ; Goldman, Daniel I.

The sandfish lizard (Scincus scincus) swims within granular media (sand) using axial body undulations to propel itself without the use of limbs. In previous work we predicted average swimming speed by developing a numerical simulation that incorporated experimentally measured biological kinematics into a multibody sandfish model. The model was coupled to an experimentally validated soft sphere discrete element method simulation of the granular medium. In this paper, we use the simulation to study the detailed mechanics of undulatory swimming in a ‘‘granular frictional fluid’’ and compare the predictions to our previously developed resistive force theory (RFT) which models sand-swimming using empirically determined granular drag laws. The simulation reveals that the forward speed of the center of mass (CoM) oscillates about its average speed in antiphase with head drag. The coupling between overall body motion and body deformation results in a non-trivial pattern in the magnitude of lateral displacement of the segments along the body. The actuator torque and segment power are maximal near the center of the body and decrease to zero toward the head and the tail. Approximately 30% of the net swimming power is dissipated in head drag. The power consumption is proportional to the frequency in the biologically relevant range, which confirms that frictional forces dominate during sand-swimming by the sandfish. Comparison of the segmental forces measured in simulation with the force on a laterally oscillating rod reveals that a granular hysteresis effect causes the overestimation of the body thrust forces in the RFT. Our models provide detailed testable predictions for biological locomotion in a granular environment.
Using blood informative transcripts in geographical genomics: impact of lifestyle on gene expression in Fijians

(Georgia Institute of Technology, 2012-11-09) Nath, Artika Praveeta ; Arafat, Dalia ; Gibson, Greg

In previous geographical genomics studies of the impact of lifestyle on gene expression inferred from microarray analysis of peripheral blood samples, we described the complex influences of culture, ethnicity, and gender in Morocco, and of pregnancy in Brisbane. Here we describe the use of nanofluidic Fluidigm quantitative RT-PCR arrays targeted at a set of 96 transcripts that are broadly informative of the major axes of immune gene expression, to explore the population structure of transcription in Fiji. As in Morocco, major differences are seen between the peripheral blood transcriptomes of rural villagers and residents of the capital city, Suva.The effect is much greater in Indian villages than in Melanesian high-landers and appears to be similar with respect to the nature of at least two axes of variation. Gender differences are much smaller than ethnicity or lifestyle effects. Body mass index is shown to associate with one of the axes as it does in Atlanta and Brisbane, establishing a link between the epidemiological transition of human metabolic disease, and gene expression profiles.
Lift-off dynamics in a simple jumping robot

(Georgia Institute of Technology, 2012-10-26) Aguilar, Jeffrey ; Lesov, Alex ; Wiesenfeld, Kurt ; Goldman, Daniel I.

We study vertical jumping in a simple robot comprising an actuated mass-spring arrangement. The actuator frequency and phase are systematically varied to find optimal performance. Optimal jumps occur above and below (but not at) the robot’s resonant frequency f0. Two distinct jumping modes emerge: a simple jump, which is optimal above f0, is achievable with a squat maneuver, and a peculiar stutter jump, which is optimal below f0, is generated with a countermovement. A simple dynamical model reveals how optimal lift-off results from nonresonant transient dynamics.
Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations

(Georgia Institute of Technology, 2012-08) Ando, Tadashi ; Chow, Edmond ; Saad, Yousef ; Skolnick, Jeffrey

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a Brownian dynamics simulation. However, the calculation of correlated Brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studies methods based on Krylov subspaces for computing Brownian noise vectors. These methods are related to Chebyshev polynomial approximations, but do not require eigenvalue estimates. We show that only low accuracy is required in the Brownian noise vectors to accurately compute values of dynamic and static properties of polymer and monodisperse suspension models. With this level of accuracy, the computational time of Krylov subspace methods scales very nearly as O(N²) for the number of particles N up to 10 000, which was the limit tested. The performance of the Krylov subspace methods, especially the “block” version, is slightly better than that of the Chebyshev method, even without taking into account the additional cost of eigenvalue estimates required by the latter. Furthermore, at N = 10 000, the Krylov subspace method is 13 times faster than the exact Cholesky method. Thus, Krylov subspace methods are recommended for performing largescale Brownian dynamics simulations with hydrodynamic interactions.