1. New encoding schemes with infofuses K.M. Park, C. Kim, S.W. Thomas III, H.J. Yoon, G. Morrison, L. Mahadevan, and G.M. Whitesides,  Advanced Matererials , 23, 4851-4856, 2011.
    [View PDF] [Download PDF] Information technology is an area of such broad importance that almost all new methodologies fi nd uses. [1] Most information derived from chemistry, and especially from some form of analysis, proceeds by a path in which a sensor (e.g., an analytical instrument) using electrical power generates information, which is then encoded and transmitted in a separate step, again using electrical power, to a separate unit, which may be local or distant, for processing and interpretation. We [2,3] and others [4–7] are exploring schemes in which information is transmitted directly, without using electrical energy. Such schemes have the potential to be useful where electrical power is not reliably available or where other constraints (e.g., size) argue for other types of solutions.
  2. Robust error correction in info fuses G. Morrison, S.W. Thomas III, C. N. LaFratta, J. Guo, M.A. Palacios, S. Sonkusale, D.R. Walt, G.M. Whitesides and L. Mahadevan,  Proceedings of the Royal Society (A) , 468, 361-377, 2011.
    [View PDF] [Download PDF] An infofuse is a combustible fuse in which information is encoded through the patterning of metallic salts, with transmission in the optical range simply associated with burning. The constraints, advantages and unique error statistics of physical chemistry require us to rethink coding and decoding schemes for these systems. We take advantage of the non-binary nature of our signal with a single bit representing one of N = 7 states to produce a code that, using a single or pair of intensity thresholds, allows the recovery of the intended signal with an arbitrarily high recovery probability, given reasonable assumptions about the distribution of errors in the system. An analysis of our experiments with infofuses shows that the code presented is consistent with these schemes, and encouraging for the field of chemical communication and infochemistry given the vast permutations and combinations of allowable nonbinary signals
  3. Shock driven jamming and periodic fracture of particulate rafts M. M. Bandi, T. Tallinen, L. Mahadevan,  Europhysics Letters Journal , 96, 36008, 2011.
    [View PDF] [Download PDF] A tenuous monolayer of hydrophobic particles at the air-water interface often forms a scum or raft. When such a monolayer is disturbed by the localized introduction of a surfactant droplet, a radially divergent surfactant shock front emanates from the surfactant origin and packs the particles into a jammed, compact, annular band with a packing fraction that saturates at a peak packing fraction φ ∗ . As the resulting two-dimensional, disordered elastic band grows with time and is driven radially outwards by the surfactant, it fractures to form periodic triangular cracks with robust geometrical features. We find that the number of cracks N and the compaction band radius R ∗ at fracture onset vary monotonically with the initial packing fraction (φinit). However, the compaction band’s width W∗ is constant for all φinit. A simple geometric theory that treats the compaction band as an elastic annulus, and accounts for mass conservation allows us to deduce that N ≃ 2πR∗ /W∗ ≃ 4πφRCP /φinit, a result we verify both experimentally and numerically. We show that the essential ingredients for this phenomenon are an initially low enough particulate packing fraction that allows surfactant-driven advection to cause passive jamming and eventual fracture of the hydrophobic particulate interface.
  4. Multistability of spontaneously curved anisotropic strips L. Giomi, L. Mahadevan,  Proceedings of the Royal Society (A) , 468, 511-530, 2011.
    [View PDF] [Download PDF] Multi-stable structures are objects with more than one stable conformation, exemplified by the simple switch. Continuum versions are often elastic composite plates or shells, such as the common measuring tape or the slap bracelet, both of which exhibit two stable configurations: rolled and unrolled. Here, we consider the energy landscape of a general class of multi-stable anisotropic strips with spontaneous Gaussian curvature. We show that while strips with non-zero Gaussian curvature can be bistable, and strips with positive spontaneous curvature are always bistable, independent of the elastic moduli, strips of spontaneous negative curvature are bistable only in the presence of spontaneous twist and when certain conditions on the relative stiffness of the strip in tension and shear are satisfied. Furthermore, anisotropic strips can become tristable when their bending rigidity is small. Our study complements and extends the theory of multi-stability in anisotropic shells and suggests new design criteria for these structures.
  5. Buckling of a thin-layer Couette flow A. Slim, J. Teichman, L. Mahadevan,  Journal of Fluid Mechanics,  1-24, 2011. 
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  6. Planar controlled gliding, tumbling and descent. P. Paoletti and L. Mahadevan,  Journal of Fluid Mechanics,  698, 489-516, 2011. 
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  7. Probability, geometry and dynamics in the toss of a thick coin. E-H. Yong, L. Mahadevan,   American Journal of Physics  79(12):1195-1201, 2011.
    [View PDF] [Download PDF] When a thick cylindrical coin is tossed in the air and lands without bouncing on an inelastic substrate, it ends up on its face or its side. We account for the rigid body dynamics of spin and precession and calculate the probability distribution of heads, tails, and sides for a thick coin as a function of its dimensions and the distribution of its initial conditions. Our theory yields a simple expression for the aspect ratio of homogeneous coins with a prescribed frequency of heads or tails compared to sides, which we validate using data from the results of tossing coins of different aspect ratios.
  8. On the growth and form of the gut T. Savin, N.A. Kurpios, A.E. Shyer, P. Florescu, H. Liang, L. Mahadevan & C.J.,  Nature  476:57-62, 2011.
    [View PDF] [Download PDF] The developing vertebrate gut tube forms a reproducible looped pattern as it grows into the body cavity. Here we use developmental experiments to eliminate alternative models and show that gut looping morphogenesis is driven by the homogeneous and isotropic forces that arise from the relative growth between the gut tube and the anchoring dorsal mesenteric sheet, tissues that grow at different rates. A simple physical mimic, using a differentially strained composite of a pliable rubber tube and a soft latex sheet is consistent with this mechanism and produces similar patterns. We devise a mathematical theory and a computational model for the number, size and shape of intestinal loops based solely on the measurable geometry, elasticity and relative growth of the tissues. The predictions of our theory are quantitatively consistent with observations of intestinal loops at different stages of development in the chick embryo. Our model also accounts for the qualitative and quantitative variation in the distinct gut looping patterns seen in a variety of species including quail, finch and mouse, illuminating how the simple macroscopic mechanics of differential growth drives the morphology of the developing gut.
  9. Dynamic instability of a growing adsorbed polymorphic filament S. Zapperi and L. Mahadevan,  Biophysical Journal  101:267-275, 2011.
    [View PDF] [Download PDF] The intermittent transition between slow growth and rapid shrinkage in polymeric assemblies is termed ‘‘dynamic instability’’, a feature observed in a variety of biochemically distinct assemblies including microtubules, actin, and their bacterial analogs. The existence of this labile phase of a polymer has many functional consequences in cytoskeletal dynamics, and its repeated appearance suggests that it is relatively easy to evolve. Here, we consider the minimal ingredients for the existence of dynamic instability by considering a single polymorphic filament that grows by binding to a substrate, undergoes a conformation change, and may unbind as a consequence of the residual strains induced by this change. We identify two parameters that control the phase space of possibilities for the filament: a structural mechanical parameter that characterizes the ratio of the bond strengths along the filament to those with the substrate (or equivalently the ratio of longitudinal to lateral interactions in an assembly), and a kinetic parameter that characterizes the ratio of timescales for growth and conformation change. In the deterministic limit, these parameters serve to demarcate a region of uninterrupted growth from that of collapse. However, in the presence of disorder in either the structural or the kinetic parameter the growth and collapse phases can coexist where the filament can grow slowly, shrink rapidly, and transition between these phases, thus exhibiting dynamic instability. We exhibit the window for the existence of dynamic instability in a phase diagram that allows us to quantify the evolvability of this labile phase.
  10. Probability, physics, and the coin toss L. Mahadevan and E-H. Yong,  Physics Today  July 2011 66-67. 
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  11. Painting with drops, jets, and sheets A. Herczynski, C. Cernuschi, and L. Mahadevan,  Physics Today  June 2011, 31-36. 
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  12. Excitable patterns in active nematics L. Giomi, L. Mahadevan, B. Chakraborty, and M. F. Hagan,  Physical Review Letters , 106, 218101, 2011.
    [View PDF] [Download PDF] We analyze a model of mutually propelled filaments suspended in a two-dimensional solvent. The system undergoes a mean-field isotropic-nematic transition for large enough filament concentrations, and the nematic order parameter is allowed to vary in space and time. We show that the interplay between nonuniform nematic order, activity, and flow results in spatially modulated relaxation oscillations, similar to those seen in excitable media. In this regime the dynamics consists of nearly stationary periods separated by ‘‘bursts’’ of activity in which the system is elastically distorted and solvent is pumped throughout. At even higher activity, the dynamics becomes chaotic
  13. The shallow turn of a worm D. Kim, S. Park, L. Mahadevan and J. Shin,  Journal of Experimental Biology , 214, 1554-1559, 2011.
    [View PDF] [Download PDF] When crawling on a solid surface, the nematode Caenorhabditis elegans (C. elegans) moves forward by propagating sinusoidal dorso-ventral retrograde contraction waves. A uniform propagating wave leads to motion that undulates about a straight line. When C. elegans turns as it forages or navigates its environment, it uses several different strategies of reorientation. These modes include the well-known omega turn, in which the worm makes a sharp angle turn forming an -shape, and the reversal, in which the worm draws itself backwards. In these two modes of reorientation, C. elegans strongly disrupts its propagating sinusoidal wave, either in form or in direction, leading to abrupt directional change. However, a third mode of reorientation, the shallow turn, involves a gentler disruption of the locomotory gait. Analyzing the statistics of locomotion suggests that the shallow turn is by far the most frequent reorienting maneuver in navigation in the absence of food. We show that the worm executes a shallow turn by modulating the amplitude and wavelength of its curvature during forward movement, and provide a minimal description of the process using a three-parameter mathematical model. The results of our study augment the understanding of how these parameters are controlled at the neuromotor circuit level.
  14. Unfolding the sulcus E. Hohlfeld and L. Mahadevan,  Physical Review Letters , 106, 105702, 2011.
    [View PDF] [Download PDF] Sulci are localized furrows on the surface of soft materials that form by a compression-induced instability. We unfold this instability by breaking its natural scale and translation invariance, and compute a limiting bifurcation diagram for sulcfication showing that it is a scale-free, subcritical nonlinear instability. In contrast with classical nucleation, sulcification is continuous, occurs in purely elastic continua and is structurally stable in the limit of vanishing surface energy. During loading, a sulcus nucleates at a point with an upper critical strain and an essential singularity in the linearized spectrum. On unloading, it quasistatically shrinks to a point with a lower critical strain, explained by breaking of scale symmetry. At intermediate strains the system is linearly stable but nonlinearly unstable with no energy barrier. Simple experiments confirm the existence of these two critical strains.
  15. How wet paper curls E. Reyssat and L. Mahadevan,  Europhysics Letters , 93, 54001, 2011.
    [View PDF] [Download PDF] When a piece of tracing paper is placed gently on the surface of a bath of water, it rapidly curls up from one edge and rolls up due to the swelling of the side in contact with water. With time, as the swelling front propagates through the thickness of the paper, the paper gradually uncurls itself and eventually straightens out. We analyze the experimental dynamics of rolling and unrolling of the paper and complement these with a minimal theory that explains the basic observations. Our study might be useful in the context of designing biomimetic devices that work as actuators or harness energy from humidity variations.
  16. Asymmetric network connectivity using weighted harmonic averages G. Morrison and L. Mahadevan,  Europhysics Letters , 93, 40002, 2011.
    [View PDF] [Download PDF] We propose a non-metric measure of the “closeness” felt between two nodes in an undirected, weighted graph using a simple weighted harmonic average of connectivity, that is a realvalued Generalized Erd¨os Number (GEN). While our measure is developed with a collaborative network in mind, the approach can be of use in a variety of artificial and real-world networks. We are able to distinguish between network topologies that standard distance metrics view as identical, and use our measure to study some simple analytically tractable networks. We show how this might be used to look at asymmetry in authorship networks such as those that inspired the integer Erd¨os numbers in mathematical coauthorships. We also show the utility of our approach to devise a ratings scheme that we apply to the data from the NetFlix prize, and find a significant improvement using our method over a baseline
  17. Growth, geometry and mechanics of the blooming lily H-Y. Liang and L. Mahadevan,  Proceedings of the National Academy of Sciences , 108, 5516-21, 2011.
    [View PDF] [Download PDF] Despite the common use of the blooming metaphor, its floral inspiration remains poorly understood. Here we study the physical process of blooming in the asiatic lily Lilium casablanca. Our observations show that the edges of the petals wrinkle as the flower opens, suggesting that differential growth drives the deployment of these laminar shell-like structures. We use a combination of surgical manipulations and quantitative measurements to confirm this hypothesis and provide a simple theory for this change in the shape of a doubly curved thin elastic shell subject to differential growth across its planform. Our experiments and theory overturn previous hypotheses that suggest that blooming is driven by differential growth of the inner layer of the petals and in the midrib by providing a qualitatively different paradigm that highlights the role of edge growth. This functional morphology suggests new biomimetic designs for deployable structures using boundary or edge actuation rather than the usual bulk or surface actuation.
  18. Structural dynamics of an actin spring L. Mahadevan, C. Riera and J. Shin  Biophysical Journal , 100, 839-44, 2011.
    [View PDF] [Download PDF] Actin-based motility in cells is usually associated with either polymerization/depolymerization in the presence of cross-linkers or contractility in the presence of myosin motors. Here, we focus on a third distinct mechanism involving actin in motility, seen in the dynamics of an active actin spring that powers the acrosomal reaction of the horseshoe crab (Limulus polyphemus) sperm. During this process, a 60-mm bent and twisted bundle of cross-linked actin uncoils and becomes straight in a few seconds in the presence of Ca2þ. This straightening, which occurs at a constant velocity, allows the acrosome to forcefully penetrate the egg. Synthesizing ultrastructural information with the kinetics, energetics, and imaging of calcium binding allows us to construct a dynamical theory for this mechanochemical engine consistent with our experimental observations. It also illuminates the general mechanism by which energy may be stored in conformational changes and released cooperatively in ordered macromolecular assemblies
  19. Elastohydrodynamics of wet bristles, carpets and brushes. A. Gopinath and L. Mahadevan,  Proceedings of the Royal Society of London (A) , 467, 1665-1685, 2011.
    [View PDF] [Download PDF] Surfaces covered by bristles, hairs, polymers and other filamentous structures arise in a variety of natural settings in science such as the active lining of many biological organs, e.g. lungs, reproductive tracts, etc., and have increasingly begun to be used in technological applications. We derive an effective field theory for the elastohydrodynamics of ordered brushes and disordered carpets that are made of a large number of elastic filaments grafted on to a substrate and interspersed in a fluid. Our formulation for the elastohydrodynamic response of these materials leads naturally to a set of constitutive equations coupling bed deformation to fluid flow, accounts for the anisotropic properties of the medium, and generalizes the theory of poroelasticity to these systems. We use the effective medium equations to study three canonical problems—the normal settling of a rigid sphere onto a carpet, the squeeze flow in a carpet and the tangential shearing motion of a rigid sphere over the carpet, all problems of relevance in mechanosensation in biology with implications for biomimetic devices.
  20. A generalized theory of viscous and inviscid flutter S. Mandre and L. Mahadevan,  Proceedings of the Royal Society of London (A) , 466, 141-156, 2010.
    [View PDF] [Download PDF] We present a unified theory of flutter in inviscid and viscous flows interacting with flexible structures based on the phenomenon of 1 : 1 resonance. We show this by treating four extreme cases corresponding to viscous and inviscid flows in confined and unconfined flows. To see the common mechanism clearly, we consider the limit when the frequencies of the first few elastic modes are closely clustered and small relative to the convective fluid time scale. This separation of time scales slaves the hydrodynamic force to the instantaneous elastic displacement and allows us to calculate explicitly the dependence of the critical flow speed for flutter on the various problem parameters. We show that the origin of the instability lies in the coincidence of the real frequencies of the first two modes at a critical flow speed beyond which the frequencies become complex, thus making the system unstable to oscillations. This critical flow speed depends on the difference between the frequencies of the first few modes and the nature of the hydrodynamic coupling between them. Our generalized framework applies to a range of elastohydrodynamic systems and further extends the Benjamin–Landahl classification of fluid–elastic instabilities.
  21. Control of shape and size of nanopillar assembly by adhesion-mediated elastocapillary interaction S. Kang, B. Pokroy, L. Mahadevan and J. Aizenberg,  ACS-Nano , 4, 11, 6323-31, 2010.
    [View PDF] [Download PDF] Control of self-organization of nanofibers into regular clusters upon evaporation-induced assembly is receiving increasing attention due to the potential importance of this process in a range of applications including particle trapping, adhesives, and structural color. Here we present a comprehensive study of this phenomenon using a periodic array of polymeric nanopillars with tunable parameters as a model system to study how geometry, mechanical properties, as well as surface properties influence capillary-induced self-organization. In particular, we show that varying the parameters of the building blocks of self-assembly provides us with a simple means of controlling the size, chirality, and anisotropy of complex structures. We observe that chiral assemblies can be generated within a narrow window for each parameter even in the absence of chiral building blocks or a chiral environment. Furthermore, introducing anisotropy in the building blocks provides a way to control both the chirality and the size of the assembly. While capillary-induced self-assembly has been studied and modeled as a quasi-static process involving the competition between only capillary and elastic forces, our results unequivocally show that both adhesion and kinetics are equally important in determining the final assembly. Our findings provide insight into how multiple parameters work together in capillary-induced self-assembly and provide us with a diverse set of options for fabricating a variety of nanostructures by self-assembly.
  22. Physiological and pathological population dynamics of circulating human red blood cells J. Higgins and L. Mahadevan,  Proceedings of the National Academy of Sciences , 107, 20587-92, 2010.
    [View PDF] [Download PDF] The systems controlling the number, size, and hemoglobin concentrations of populations of human red blood cells (RBCs), and their dysregulation in anemia, are poorly understood. After release from the bone marrow, RBCs undergo reduction in both volume and total hemoglobin content by an unknown mechanism [Lew VL, et al. (1995) Blood 86:334–341; Waugh RE, et al. (1992) Blood 79:1351–1358]; after ∼120 d, responding to an unknown trigger, they are removed. We used theory from statistical physics and data from the hospital clinical laboratory [d’Onofrio G, et al. (1995) Blood 85:818–823] to develop a master equation model for RBC maturation and clearance. The model accurately identifies patients with anemia and distinguishes thalassemia-trait anemia from irondeficiency anemia. Strikingly, it also identifies many pre-anemic patients several weeks before anemia becomes clinically detectable. More generally we illustrate how clinical laboratory data can be used to develop and to test a dynamic model of human pathophysiology with potential clinical utility.
  23. Why subduction zones are curved L. Mahadevan, R. Bendick and H-Y. Liang,  Tectonics , 29,TC6002, 2010.
    [View PDF] [Download PDF] We give an explanation for the polarity, localization, shape, size, and initiation of subduction zones on Earth. By considering a soft, thin, curved lithospheric cap with either elastic or viscous rheology supported by a thick, nearly incompressible mantle, we find two different characteristic subduction geometries arise depending on boundary conditions: (1) plate boundaries where subduction results primarily from the gravitational body force (free subduction) have characteristic plate lengths and form arc‐shaped dimpled segments resulting from the competition between bending and stretching in edge buckling modes of thin spherical shells, and (2) subduction zones due to localized applied loads that push one slab of thin, positively buoyant lithosphere beneath an overriding plate (forced subduction) form localized straight segments, consistent with the deformation of indented spherical shells. Both types of subduction are nonlinear subcritical instabilities, so small perturbations in the mechanical properties of the lithosphere have pronounced effects on subduction initiation and evolution. Yet in both cases, geometric relationships determined by the shape of the Earth itself play the most critical role in controlling the basic morphology and characteristic length scales of subduction zones
  24. Geometry, mechanics and electronics of singular structures in graphene V. Pereira, H-Y Liang, A. Castro-Neto and L. Mahadevan,  Physical Review Letters , 105, 156603, 2010.
    [View PDF] [Download PDF] As the thinnest atomic membrane, graphene presents an opportunity to combine geometry, elasticity, and electronics at the limits of their validity. We describe the transport and electronic structure in the neighborhood of conical singularities, the elementary excitations of the ubiquitous wrinkled and crumpled graphene. We use a combination of atomistic mechanical simulations, analytical geometry, and transport calculations in curved graphene, and exact diagonalization of the electronic spectrum to calculate the effects of geometry on electronic structure, transport, and mobility in suspended samples, and how the geometry-generated pseudomagnetic and pseudoelectric fields might disrupt Landau quantization.
  25. Cooperative adhesion and friction of compliant nanohairs  L. Ge, A. Goyal, R. Shi, L. Mahadevan, P. Ajayan and A. Dhinojwala,  Nanoletters , 10, 4509, 2010.
    [View PDF] [Download PDF] The adhesion and friction behavior of soft materials, including compliant brushes and hairs, depends on the temporal and spatial evolution of the interfaces in contact. For compliant nanofibrous materials, the actual contact area individual fibers make with surfaces depends on the preload applied upon contact. Using in situ microscopy observations of preloaded nanotube hairs, we show how nanotubes make cooperative contact with a surface by buckling and conforming to the surface topography. The overall adhesion of compliant nanohairs increases with increasing preload as nanotubes deform and continuously add new side-wall contacts with the surface. Electrical resistance measurements indicate significant hysteresis in the relative contact area. Contact area increases with preload (or stress) and decreases suddenly during unloading, consistent with strong adhesion observed for these complaint nanohairs.
  26. Nanopottery: coiling of electrospun polymer nanofibers H-Y. Kim, M. Lee, K-J. Park, S. Kim and L. Mahadevan,  Nanoletters , 10, 2138, 2010.
    [View PDF] [Download PDF] The buckling, folding, and coiling of thin sheets and filaments of solids and fluids take place on length scales spanning several orders of magnitude, in phenomena ranging from orogenesis in geophysics to materials processing and soft-matter physics. For example, when an elastic rope is fed uniformly toward a horizontal plane, it first buckles and eventually coils into a spool that is deposited onto the plane.1 A similar phenomenon also occurs when a slender viscous fluid jet impinges onto a horizontal plane and leads to the deposition of a liquid rope coil.2 In either case, although the scale of the coil and the speed of coiling are determined by the balance between the internal elastic or viscous forces that resist deformation and a combination of inertia and gravity, the basic phenomenology is a consequence of geometry which favors bending deformations over stretching modes. Here, we consider the spon
  27. Statistical mechanics of developable ribbons L. Giomi and L. Mahadevan,  Physical Review Letters , 104, 238104, 2010.
    [View PDF] [Download PDF] We investigate the statistical mechanics of long developable ribbons of finite width and very small thickness. The constraint of isometric deformations in these ribbonlike structures that follows from the geometric separation of scales introduces a coupling between bending and torsional degrees of freedom. Using analytical techniques and Monte Carlo simulations, we find that the tangent-tangent correlation functions always exhibit an oscillatory decay at any finite temperature implying the existence of an underlying helical structure even in the absence of a preferential zero-temperature twist. In addition, the persistence length is found to be over 3 times larger than that of a wormlike chain having the same bending rigidity. Our results are applicable to many ribbonlike objects in polymer physics and nanoscience that cannot be described by the classical wormlike chain model.
  28. Influence of feedback on the stochastic evolution of simple climate systems L. Mahadevan and J. Deutch,  Proceedings of the Royal Society of London (A) , 466, 993, 2010.
    [View PDF] [Download PDF] We consider the dynamical evolution of a simple climate system that describes the average temperature of the Earth’s atmosphere owing to radiative forcing and coupling to a positive feedback variable such as the concentration of greenhouse gases in the presence of fluctuations. Analysing the resulting stochastic dynamical system shows that, if the temperature relaxes rapidly relative to the concentration, the time-dependent and stationary probability density functions (pdfs) for the temperature rise possess a fat tail. In contrast, if the feedback variable relaxes rapidly relative to the temperature, the pdf has no fat tail, and, instead, the system shows critical slowing down as the singular limit of positive feedback is approached. However, if there is uncertainty in the feedback variable itself, a fat tail can reappear. Our analysis may be generalized to more complex models with similar qualitative results. Our results have policy implications: although fat tails imply that the expectation of plausible damage functions is infinite, the pdfs permit an examination of the trade-off between reducing emissions and reducing the positive feedback gain.
  29. Shape and dynamics of tip-growing cells O. Campas and L. Mahadevan,  Current Biology , 19, 2102, 2009.
    [View PDF] [Download PDF] Walled cells have the ability to remodel their shape while sustaining an internal turgor pressure that can reach values up to 10 atmospheres [1–7]. Although it is undisputed that this requires a tight and simultaneous regulation of cell wall assembly and mechanics, previous theoretical studies on tip growth focused either on the mechanical behavior of the cell wall or on its assembly [8–14]. To study the interplay between growth and mechanics in shaping a walled cell, we examine the particularly simple geometry of tip-growing cells [1, 3, 15, 16], which elongate via the assembly and expansion of cell wall in the apical region of the cell. We describe the observed irreversible expansion of the cell wall during growth as the extension of an inhomogeneous viscous fluid shell under the action of turgor pressure, fed by a material source in the neighborhood of the growing tip. This allows us to determine theoretically the radius of the cell and its growth velocity in terms of the turgor pressure and the secretion rate and rheology of the cell wall material. We derive simple scaling laws for the geometry of the cell and find that a single dimensionless parameter, which characterizes the relative roles of cell wall assembly and expansion, is sufficient to explain the observed variability in shapes of tip-growing cells. More generally, our description provides a framework to understand cell growth and remodeling in plants (pollen tubes [17], root hairs, etc. [18]), fungi (hyphal growth [19, 20] and fission and budding yeast [3]), and some bacteria [21], in the context of both tip growth and diffuse growth.
  30. The shape of a long leaf H. Liang and L. Mahadevan,  Proceedings of the National Academy of Sciences (USA) ,106, 22049, 2009.
    [View PDF] [Download PDF] Long leaves in terrestrial plants and their submarine counterparts, algal blades, have a typical, saddle-like midsurface and rippled edges. To understand the origin of these morphologies, we dissect leaves and differentially stretch foam ribbons to show that these shapes arise from a simple cause, the elastic relaxation via bending that follows either differential growth (in leaves) or differential stretching past the yield point (in ribbons). We quantify these different modalities in terms of a mathematical model for the shape of an initially flat elastic sheet with lateral gradients in longitudinal growth. By using a combination of scaling concepts, stability analysis, and numerical simulations, we map out the shape space for these growing ribbons and find that as the relative growth strain is increased, a long flat lamina deforms to a saddle shape and/or develops undulations that may lead to strongly localized ripples as the growth strain is localized to the edge of the leaf. Our theory delineates the geometric and growth control parameters that determine the shape space of finite laminae and thus allows for a comparative study of elongated leaf morphology.
  31. Flip-flop-induced relaxation of bending energy: implications for membrane remodeling R. Bruckner, S. Mansy, A. Ricardo, L. Mahadevan, and J. Szostak,  Biophysical Journal , 97, 3113, 2009
    [View PDF] [Download PDF] Cellular and organellar membranes are dynamic materials that underlie many aspects of cell biology. Biological membranes have long been thought of as elastic materials with respect to bending deformations. A wealth of theory and experimentation on pure phospholipid membranes provides abundant support for this idea. However, biological membranes are not composed solely of phospholipids—they also incorporate a variety of amphiphilic molecules that undergo rapid transbilayer flip-flop. Here we describe several experimental systems that demonstrate deformation-induced molecular flip-flop. First we use a fluorescence assay to track osmotically controlled membrane deformation in single component fatty acid vesicles, and show that the relaxation of the induced bending stress is mediated by fatty acid flip-flop. We then look at two-component phospholipid/cholesterol composite vesicles. We use NMR to show that the steady-state rate of interleaflet diffusion of cholesterol is fast relative to biological membrane remodeling. We then use a Fo¨rster resonance energy transfer assay to detect the transbilayer movement of cholesterol upon deformation. We suggest that our results can be interpreted by modifying the area difference elasticity model to account for the time-dependent relaxation of bending energy. Our findings suggest that rapid interleaflet diffusion of cholesterol may play a role in membrane remodeling in vivo. We suggest that the molecular characteristics of sterols make them evolutionarily preferred mediators of stress relaxation, and that the universal presence of sterols in the membranes of eukaryotes, even at low concentrations, reflects the importance of membrane remodeling in eukaryotic cells.
  32. Controlling the orientation and synaptic differentiation of myotubes with micropatterned substrates J. Gingras, R. Rioux, D. Cuvelier, N. Geisse, J. Lichtman, G. Whitesides, L. Mahadevan, and J. Sanes,  Biophysical Journa l, 97, 2771, 2009.
    [View PDF] [Download PDF] Micropatterned poly(dimethylsiloxane) substrates fabricated by soft lithography led to large-scale orientation of myoblasts in culture, thereby controlling the orientation of the myotubes they formed. Fusion occurred on many chemically identical surfaces in which varying structures were arranged in square or hexagonal lattices, but only a subset of patterned surfaces yielded aligned myotubes. Remarkably, on some substrates, large populations of myotubes oriented at a reproducible acute angle to the lattice of patterned features. A simple geometrical model predicts the angle and extent of orientation based on maximizing the contact area between the myoblasts and patterned topographic surfaces. Micropatterned substrates also provided short-range cues that influenced higher-order functions such as the localization of focal adhesions and accumulation of postsynaptic acetylcholine receptors. Our results represent what we believe is a new approach for musculoskeletal tissue engineering, and our model sheds light on mechanisms of myotube alignment in vivo.
  33. The shape and motion of a ruck in a rug J. Kolinski, P. Aussillous and L. Mahadevan,  Physical Review Letters , 103, 174302, 2009.
    [View PDF] [Download PDF] The motion of a ruck in a rug is used as an analogy to explain the role of dislocations in crystalline solids. We take literally one side of this analogy and study the shape and motion of a bump, wrinkle or ruck in a thin sheet in partial contact with a rough substrate in a gravitational field. Using a combination of experiments, scaling analysis and numerical solutions of the governing equations, we quantify the static shape of a ruck on a horizontal plane. When the plane is inclined, the ruck becomes asymmetric and moves by rolling only when the inclination of the plane reaches a critical angle, at a speed determined by a simple power balance. We find that the angle at which rolling starts is larger than the angle at which the ruck stops; i.e., static rolling friction is larger than dynamic rolling friction. We conclude with a generalization of our results to wrinkles in soft adherent extensible films.
  34. Infochemistry: encoding information as optical pulses using droplets in a microfluidic device M. Hashimoto, J. Feng, R. York, A. Ellerbee, G. Morrison, S. Thomas,  L. Mahadevan, and G. Whitesides,  Journal of the American Chemical Society , 131, 12420, 2009.
    [View PDF] [Download PDF] This article describes a new procedure for generating and transmitting a messagesa sequence of optical pulsessby aligning a mask (an opaque sheet containing transparent “windows”) below a microfluidic channel in which flows an opaque continuous fluid containing transparent droplets. The optical mask encodes the message as a unique sequence of windows that can transmit or block light; the flow of transparent droplets in the channel converts this message into a sequence of optical pulses. The properties of the windows on the mask (e.g., their size, wavelength of transmittance, orientation of polarization) determine the information carried in these optical pulses (e.g., intensity, color, polarization). The structure of a transmitted signal depends on the number and spacing of droplets in the channel. Fourier transformation can deconvolve superimposed signals created by the flow of multiple droplets into the message that a single droplet would transmit. The research described in this contribution explores a new field at the intersection of chemistry, materials science, and information technology: infochemistry.
  35. Animal cell hydraulics G. Charras, T. Mitchison and L. Mahadevan,  Journal of Cell Science , 122, 3233, 2009.
    [View PDF] [Download PDF] Water is the dominant ingredient of cells and its dynamics are crucial to life. We and others have suggested a physical picture of the cell as a soft, fluid-infiltrated sponge, surrounded by a water-permeable barrier. To understand water movements in an animal cell, we imposed an external, inhomogeneous osmotic stress on cultured cancer cells. This forced water through the membrane on one side, and out on the other. Inside the cell, it created a gradient in hydration, that we visualized by tracking cellular responses using natural organelles and artificially introduced quantum dots. The dynamics of these markers at short times were the same for normal and metabolically poisoned cells, indicating that the cellular responses are primarily physical rather than chemical. Our finding of an internal gradient in hydration is inconsistent with a continuum model for cytoplasm, but consistent with the sponge model, and implies that the effective pore size of the sponge is small enough to retard water flow significantly on time scales (~10–100 seconds) relevant to cell physiology. We interpret these data in terms of a theoretical framework that combines mechanics and hydraulics in a multiphase poroelastic description of the cytoplasm and explains the experimentally observed dynamics quantitatively in terms of a few coarse-grained parameters that are based on microscopically measurable structural, hydraulic and mechanical properties. Our fluid-filled sponge model could provide a unified framework to understand a number of disparate observations in cell morphology and motility.
  36. Calcium regulation of an actin spring B. Tam, J. Shin, E. Pfeiffer, P. Matsudaira and L. Mahadevan,  Biophysical Journal , 97, 1125, 2009.
    [View PDF] [Download PDF] Calcium is essential for many biological processes involved in cellular motility. However, the pathway by which calcium influences motility, in processes such as muscle contraction and neuronal growth, is often indirect and complex. We establish a simple and direct mechanochemical link that shows how calcium quantitatively regulates the dynamics of a primitive motile system, the actin-based acrosomal bundle of horseshoe crab sperm. The extension of this bundle requires the continuous presence of external calcium. Furthermore, the extension rate increases with calcium concentration, but at a given concentration, we find that the volumetric rate of extension is constant. Our experiments and theory suggest that calcium sequentially binds to calmodulin molecules decorating the actin filaments. This binding leads to a collective wave of untwisting of the actin filaments that drives bundle extension.
  37. Strain induced alignment in collagen gels D. Vader, A. Kabla, D. Weitz and L. Mahadevan,  PLoS One , 4 (6), e5902, 2009.
    [View PDF] [Download PDF] Collagen is the most abundant extracellular-network-forming protein in animal biology and is important in both natural and artificial tissues, where it serves as a material of great mechanical versatility. This versatility arises from its almost unique ability to remodel under applied loads into anisotropic and inhomogeneous structures. To explore the origins of this property, we develop a set of analysis tools and a novel experimental setup that probes the mechanical response of fibrous networks in a geometry that mimics a typical deformation profile imposed by cells in vivo. We observe strong fiber alignment and densification as a function of applied strain for both uncrosslinked and crosslinked collagenous networks. This alignment is found to be irreversibly imprinted in uncrosslinked collagen networks, suggesting a simple mechanism for tissue organization at the microscale. However, crosslinked networks display similar fiber alignment and the same geometrical properties as uncrosslinked gels, but with full reversibility. Plasticity is therefore not required to align fibers. On the contrary, our data show that this effect is part of the fundamental non-linear properties of fibrous biological networks.
  38. Cell shape changes indicate a role for extrinsic tensile forces in Drosophila germ-band extension L. Butler, G. Blanchard, A. Kabla, N. Lawrence, D. Welchman, L. Mahadevan, R. Adams and B. Sanson,  Nature Cell Biology , 11(7), 859-64, 2009.
    [View PDF] [Download PDF] Drosophila germ-band extension (GBE) is an example of the convergence and extension movements that elongate and narrow embryonic tissues. To understand the collective cell behaviours underlying tissue morphogenesis, we have continuously quantified cell intercalation and cell shape change during GBE. We show that the fast, early phase of GBE depends on cell shape change in addition to cell intercalation. In antero-posterior patterning mutants such as those for the gap gene Krüppel, defective polarized cell intercalation is compensated for by an increase in antero-posterior cell elongation, such that the initial rate of extension remains the same. Spatio-temporal patterns of cell behaviours indicate that an antero-posterior tensile force deforms the germ band, causing the cells to change shape passively. The rate of anteroposterior cell elongation is reduced in twist mutant embryos, which lack mesoderm. We propose that cell shape change contributing to germ-band extension is a passive response to mechanical forces caused by the invaginating mesoderm.
  39. Tissue Tectonics: morphogenetic strain rates, cell shape change and intercalation G. Blanchard, A. Kabla, L. Butler, B. Sanson, N. Gorfinkiel, L. Mahadevan and R. Adams,  Nature Methods , 6(6), 458-64, 2009.
    [View PDF] [Download PDF] The dynamic reshaping of tissues during morphogenesis results from a combination of individual cell behaviors and collective cell rearrangements. However, a comprehensive framework to unambiguously measure and link cell behavior to tissue morphogenesis is lacking. Here we introduce such a kinematic framework, bridging cell and tissue behaviors at an intermediate, mesoscopic, level of cell clusters or domains. By measuring domain deformation in terms of the relative motion of cell positions and the evolution of their shapes, we characterized the basic invariant quantities that measure fundamental classes of cell behavior, namely tensorial rates of cell shape change and cell intercalation. In doing so we introduce an explicit definition of cell intercalation as a continuous process. We mapped strain rates spatiotemporally in three models of tissue morphogenesis, gaining insight into morphogenetic mechanisms. Our quantitative approach has broad relevance for the precise characterization and comparison of morphogenetic phenotypes.
  40. Hygromorphs: from pine cones to biomimetic bilayers E. Reyssat and L. Mahadevan,  Journal of the Royal Society, Interface , 6, 951-957, 2009.
    [View PDF] [Download PDF] We consider natural and artificial hygromorphs, objects that respond to environmental humidity by changing their shape. Using the pine cone as an example that opens when dried and closes when wet, we quantify the geometry, mechanics and dynamics of closure and opening at the cell, tissue and organ levels, building on our prior structural knowledge. A simple scaling theory allows us to quantify the hysteretic dynamics of opening and closing. We also show how simple bilayer hygromorphs of paper and polymer show similar behaviour that can be quantified via a theory which couples fluid transport in a porous medium and evaporative flux to mechanics and geometry. Our work unifies varied observations of natural hygromorphs and suggests interesting biomimetic analogues, which we illustrate using an artificial flower with a controllable blooming and closing response.
  41. Botanical ratchets I. Kulic, M. Mani, H. Mohrbach, R. Thaokar, L. Mahadevan,  Proceedings of the Royal Society of London (B), Biological Sciences , 276, 2243-47, 2009.
    [View PDF] [Download PDF] Ratcheting surfaces are a common motif in nature and appear in plant awns and grasses. They are known to proffer selective advantages for seed dispersion and burial. In two simple model experiments, we show that these anisotropically toothed surfaces naturally serve as motion rectifiers and generically move in a unidirectional manner, when subjected to temporally and spatially symmetric excitations of various origins. Using a combination of theory and experiment, we show that a linear relationship between awn length and ratchet efficiency holds under biologically relevant conditions. Grass awns can thus efficiently transform non-equilibrium environmental stresses from such sources as humidity variations into useful work and directed motion using their length as a fluctuation amplifier, yielding a selective advantage to these organelles in many plant species.
  42. Statistical dynamics of flowing red blood cells by morphological image processing J. Higgins, D. Eddington, S. Bhatia and L. Mahadevan,  PLoS Computational Biology , 5, e1000288, 2009.
    [View PDF] [Download PDF] Blood is a dense suspension of soft non-Brownian cells of unique importance. Physiological blood flow involves complex interactions of blood cells with each other and with the environment due to the combined effects of varying cell concentration, cell morphology, cell rheology, and confinement. We analyze these interactions using computational morphological image analysis and machine learning algorithms to quantify the non-equilibrium fluctuations of cellular velocities in a minimal, quasi-two-dimensional microfluidic setting that enables high-resolution spatio-temporal measurements of blood cell flow. In particular, we measure the effective hydrodynamic diffusivity of blood cells and analyze its relationship to macroscopic properties such as bulk flow velocity and density. We also use the effective suspension temperature to distinguish the flow of normal red blood cells and pathological sickled red blood cells and suggest that this temperature may help to characterize the propensity for stasis in Virchow’s Triad of blood clotting and thrombosis.
  43. Self-organization of a mesoscale bristle into ordered hierarchical helical assemblies B. Pokroy, S. Kang, L. Mahadevan, and J. Aizenberg,  Science , 323, 237, 2009.
    [View PDF] [Download PDF] Mesoscale hierarchical helical structures with diverse functions are abundant in nature. Here we show how spontaneous helicity can be induced in a synthetic polymeric nanobristle assembling in an evaporating liquid. We use a simple theoretical model to characterize the geometry, stiffness, and surface properties of the pillars that favor the adhesive self-organization of bundles with pillars wound around each other. The process can be controlled to yield highly ordered helical clusters with a unique structural hierarchy that arises from the sequential assembly of self-similar coiled building blocks over multiple length scales. We demonstrate their function in the context of self-assembly into previously unseen structures with uniform, periodic patterns and controlled handedness and as an efficient particle-trapping and adhesive system.
  44. Nonequilibrium scale selection mechanism for columnar jointing L. Goehring L. Mahadevan, and S. Morris,  Proceedings of the National Academy of Sciences (USA) , 106, 387, 2009.
    [View PDF] [Download PDF] Crack patterns in laboratory experiments on thick samples of drying cornstarch are geometrically similar to columnar joints in cooling lava found at geological sites such as the Giant’s Causeway. We present measurements of the crack spacing from both laboratory and geological investigations of columnar jointing, and show how these data can be collapsed onto a single master scaling curve. This is due to the underlying mathematical similarity between theories for the cracking of solids induced by differential drying or by cooling. We use this theory to give a simple quantitative explanation of how these geometrically similar crack patterns arise from a single dynamical law rooted in the nonequilibrium nature of the phenomena. We also give scaling relations for the characteristic crack spacing in other limits consistent with our experiments and observations, and discuss the implications of our results for the control of crack patterns in thin and thick solid films.
  45. Elasticity of floppy and stiff random networks M. Wyart, H. Liang, A. Kabla,and L. Mahadevan,  Physical Review Letters , 101, 215501, 2008.
    [View PDF] [Download PDF] We study the linear and nonlinear elastic behavior of amorphous systems using a two-dimensional random network of harmonic springs as a model system. A natural characterization of these systems arises in terms of the network coordination (average number of springs per node) relative to that of a marginally rigid network z: a floppy network has z < 0, while a stiff network has z > 0. Under the influence of an externally applied load, we observe that the response of both floppy and stiff networks is controlled by the critical point corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the heterogeneity of the response, and the network stiffening as a function of z and derive these theoretically, thus allowing us to predict aspects of the mechanical response of glasses and fibrous networks.
  46. Quantifying the relation between bond number and myoblast proliferation T. Boontheekul, Hyun-Joon Kong, S. Hsiong, Y-C. Huang, L. Mahadevan, H. Vandenburgh and D. Mooney,  Faraday Discussions , 139, 53, 2008.
    [View PDF] [Download PDF] Many functions of the extracellular matrix can be mimicked by small peptide fragments (e.g., arginine–glycine–aspartic acid (RGD) sequence) of the entire molecule, but the presentation of the peptides is critical to their effects on cells. It is likely that some effects of peptide presentation from biomaterials simply relate to the number of bonds formed between cell receptors and the adhesion ligands, but a lack of tools to quantify bond number limits direct investigation of this assumption. The impact of different ligand presentations (density, affinity, and nanoscale distribution) on the proliferation of C2C12 and human primary myoblasts was first examined in this study. Increasing the ligand density or binding affinity led to a similar enhancement in proliferation of C2C12 cells and human primary myoblasts. The nanoscale distribution of clustered RGD ligands also influenced C2C12 cells and human primary myoblast proliferation, but in an opposing manner. A rheological technique and a FRET technique were then utilized to quantify the number of receptor–ligand interactions as a function of peptide presentation. Higher numbers of bonds were formed when the RGD density and affinity were increased, as measured with both techniques, and bond number correlated with cell growth rates. However, the influence of the nanoscale peptide distribution did not appear to be solely a function of bond number. Altogether, these findings provide significant insight to the role of peptide presentation in the regulation of cell proliferation, and the approaches developed in this work may have significant utility in probing how adhesion regulates a variety of other cellular functions and aid in developing design criterion for cell-interactive materials.
  47. Polymer science and biology: structure and dynamics at multiple scales
    L. Mahadevan,  Faraday Discussions , 139, 9, 2008. 
    [View PDF] [Download PDF]
  48. Implications of a poroelastic cytoplasm for the dynamics of animal cell shape T. Mitchison, G. Charras and L. Mahadevan,  Seminars in Cell & Developmental Biology  19, 215, 2008.
    [View PDF] [Download PDF] Two views have dominated recent discussions of the physical basis of cell shape change during migration and division of animal cells: the cytoplasm can be modeled as a viscoelastic continuum, and the forces that change its shape are generated only by actin polymerization and actomyosin contractility in the cell cortex. Here, we question both views: we suggest that the cytoplasm is better described as poroelastic, and that hydrodynamic forces may be generally important for its shape dynamics. In the poroelastic view, the cytoplasm consists of a porous, elastic solid (cytoskeleton, organelles, ribosomes) penetrated by an interstitial fluid (cytosol) that moves through the pores in response to pressure gradients. If the pore size is small (30–60 nm), as has been observed in some cells, pressure does not globally equilibrate on time and length scales relevant to cell motility. Pressure differences across the plasma membrane drive blebbing, and potentially other type of protrusive motility. In the poroelastic view, these pressures can be higher in one part of a cell than another, and can thus cause local shape change. Local pressure transients could be generated by actomyosin contractility, or by local activation of osmogenic ion transporters in the plasma membrane. We propose that local activation of Na+/H+ antiporters (NHE1) at the front of migrating cells promotes local swelling there to help drive protrusive motility, acting in combination with actin polymerization. Local shrinking at the equator of dividing cells may similarly help drive invagination during cytokinesis, acting in combination with actomyosin contractility. Testing these hypotheses is not easy, as water is a difficult analyte to track, and will require a joint effort of the cytoskeleton and ion physiology communities.
  49. How kelp produce blade shapes suited to different flow regimes: A new wrinkle M. Koehl, W. Silk, H. Liang and L. Mahadevan,  Integrative and Comparative Biology , 48(6), 834, 2008.
    [View PDF] [Download PDF] Synopsis Many species of macroalgae have flat, strap-like blades in habitats exposed to rapidly flowing water, but have wide, ruffled ‘‘undulate’’ blades at protected sites. We used the giant bull kelp, Nereocystis luetkeana, to investigate how these ecomorphological differences are produced. The undulate blades of N. luetkeana from sites with low flow remain spread out and flutter erratically in moving water, thereby not only enhancing interception of light, but also increasing drag. In contrast, strap-like blades of kelp from habitats with rapid flow collapse into streamlined bundles and flutter at low amplitude in flowing water, thus reducing both drag and interception of light. Transplant experiments in the field revealed that shape of the blade in N. luetkeana is a plastic trait. Laboratory experiments in which growing blades from different sites were subjected to tensile forces that mimicked the hydrodynamic drag experienced by blades in different flow regimes showed that change in shape is induced by mechanical stress. During growth experiments in the field and laboratory, we mapped the spatial distribution of growth in both undulate and strap-like blades to determine how these different morphologies were produced. The highest growth rates occur near the proximal ends of N. luetkeana blades of both morphologies, but the rates of transverse growth of narrow, strap-like blades are lower than those of wide, undulate blades. If rates of longitudinal growth at the edges of a blade exceed the rate of longitudinal growth along the midline of the blade, ruffles along the edges of the blade are produced by elastic buckling. In contrast, flat blades are produced when rates of longitudinal growth are similar across the width of a blade. Because ruffles are the result of elastic buckling, a compliant undulate N. luetkeana blade can easily be pushed into different configurations (e.g., the wavelengths of the ruffles along the edges of the blade can change, and the whole blade can twist into left- and right-handed helicoidal shapes), which may enhance movements of the blade in flowing water that reduce self-shading and increase mass exchange along blade surfaces
  50. Dynamics of chromatin decondensation reveals the structural integrity of a mechanically prestressed nucleus  A. Mazumder, T. Roopa, A. Basu, L. Mahadevan, and G. Shivashankar,  Biophysical Journal , 95, 3028, 2008.
    [View PDF] [Download PDF] Genome organization within the cell nucleus is a result of chromatin condensation achieved by histone tail-tail interactions and other nuclear proteins that counter the outward entropic pressure of the polymeric DNA. We probed the entropic swelling of chromatin driven by enzymatic disruption of these interactions in isolated mammalian cell nuclei. The large-scale decondensation of chromatin and the eventual rupture of the nuclear membrane and lamin network due to this entropic pressure were observed by fluorescence imaging. This swelling was accompanied by nuclear softening, an effect that we quantified by measuring the fluctuations of an optically trapped bead adhered onto the nucleus. We also measured the pressure at which the nuclear scaffold ruptured using an atomic force microscope cantilever. A simple theory based on a balance of forces in a swelling porous gel quantitatively explains the diffusive dynamics of swelling. Our experiments on decondensation of chromatin in nuclei suggest that its compaction is a critical parameter in controlling nuclear stability
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