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This work quantified how cell velocity and effective power output are influenced by the rate of actin depolymerization, which is affected by myosin contraction. Model analysis shows that the cell migration velocity displays a biphasic dependence on the rate of actin depolymerization and myosin contraction.

The bidomain model is the standard model for cardiac electrophysiology. In this paper, we investigate the instability and asymptotic behavior of planar fronts and planar pulses of the bidomain Allen-Cahn equation and the bidomain FitzHugh-Nagumo equation in two spatial dimension. In previous work, it was shown that planar fronts of the bidomain Allen-Cahn equation can become unstable in contrast to the classical Allen-Cahn equation. We find that, after the planar front is destabilized, a rotating zigzag front develops whose shape can be explained by simple geometric arguments using a suitable Frank diagram. We also show that the Hopf bifurcation through which the front becomes unstable can be either supercritical or subcritical, by demonstrating a parameter regime in which a stable planar front and zigzag front can coexist. In our computational studies of the bidomain FitzHugh-Nagumo pulse solution, we show that the pulses can also become unstable much like the bidomain Allen-Cahn fronts. However, unlike the bidomain Allen-Cahn case, the destabilized pulse does not necessarily develop into a zigzag pulse. For certain choice of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by the development of a numerical scheme that allows for the accurate computation of the bidomain equation in a two dimensional strip domain of infinite extent., Comment: 29 pages

Human societies include diverse social relationships. Friends, family, business colleagues, and online contacts can all contribute to one's social life. Individuals may behave differently in different domains, but success in one domain may engender success in another. Here, we study this problem using multilayer networks to model multiple domains of social interactions, in which individuals experience different environments and may express different behaviors. We provide a mathematical analysis and find that coupling between layers tends to promote prosocial behavior. Even if prosociality is disfavored in each layer alone, multilayer coupling can promote its proliferation in all layers simultaneously. We apply this analysis to six real-world multilayer networks, ranging from the socio-emotional and professional relationships in a Zambian community, to the online and offline relationships within an academic University. We discuss the implications of our results, which suggest that small modifications to interactions in one domain may catalyze prosociality in a different domain., 68 pages; final version

We consider the accuracy of slender body theory in approximating the force exerted by a thin fiber on the surrounding viscous fluid when the fiber velocity is prescribed. We term this the slender body inverse problem, as it is known that slender body theory converges to a well-posed PDE solution when the force is prescribed and the fiber velocity is unknown. From a PDE perspective, the slender body inverse problem is simply the Dirichlet problem for the Stokes equations, but from an approximation perspective, nonlocal slender body theory exhibits high wavenumber instabilities which complicate analysis. Here we consider two methods for regularizing the slender body approximation: spectral truncation and the $\delta$-regularization of Tornberg and Shelley (2004). For a straight, periodic fiber with constant radius $\epsilon>0$, we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We show that the spectrum of the original slender body approximation agrees closely with the PDE solution at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff $\sim1/\epsilon$. For both the truncated and $\delta$-regularized approximations, we obtain similar convergence results to the PDE solution as $\epsilon\to0$: a fiber velocity with $H^1$ regularity gives $O(\epsilon)$ convergence, while a fiber velocity with at least $H^2$ regularity yields $O(\epsilon^2)$ convergence. Moreover, we determine the dependence of the $\delta$-regularized error estimate on the regularization parameter $\delta$.

We consider a classical elastohydrodynamic model of an inextensible filament undergoing planar motion in $\mathbb{R}^3$. The hydrodynamics are described by resistive force theory, and the fiber elasticity is governed by Euler-Bernoulli beam theory. Our aim is twofold: (1) Serve as a starting point for developing the mathematical analysis of filament elastohydrodynamics, particularly the analytical treatment of an inextensibility constraint, and (2) As an application, prove conditions on internal fiber forcing that allow a free-ended filament to swim. Our analysis of fiber swimming speed is supplemented with a numerical optimization of the internal fiber forcing, as well as a novel numerical method for simulating an inextensible swimmer., Comment: 38 pages, many figures toward the end. Some typos fixed

Biological and social systems are structured at multiple scales, and the incentives of individuals who interact in a group may diverge from the collective incentive of the group as a whole. Mechanisms to resolve this tension are responsible for profound transitions in evolutionary history, including the origin of cellular life, multi-cellular life, and even societies. Here we synthesize a growing literature that extends evolutionary game theory to describe multilevel evolutionary dynamics, using nested birth-death processes and partial differential equations to model natural selection acting on competition within and among groups of individuals. We apply this theory to analyze how mechanisms known to promote cooperation within a single group -- including assortment, reciprocity, and population structure -- alter evolutionary outcomes in the presence of competition among groups. We find that population structures most conducive to cooperation in multi-scale systems may differ from those most conducive within a single group. Likewise, for competitive interactions with a continuous range of strategies we find that among-group selection may fail to produce socially optimal outcomes, but it can nonetheless produce second-best solutions that balance individual incentives to defect with the collective incentives for cooperation. We conclude by describing the broad applicability of multi-scale evolutionary models to problems ranging from the production of diffusible metabolites in microbes to the management of common-pool resources in human societies., Comment: 48 pages, 8 figures

Intercellular signaling has an important role in organism development, but not all communication occurs using the same mechanism. Here, we analyze the energy efficiency of intercellular signaling by two canonical mechanisms: diffusion of signaling molecules and direct transport mediated by signaling cellular protrusions. We show that efficient contact formation for direct transport can be established by an optimal rate of projecting protrusions, which depends on the availability of information about the location of the target cell. The optimal projection rate also depends on how signaling molecules are transported along the protrusion, in particular the ratio of the energy cost for contact formation and molecule synthesis. Also, we compare the efficiency of the two signaling mechanisms, under various model parameters. We find that the direct transport is favored over the diffusion when transporting a large amount of signaling molecules. There is a critical number of signaling molecules at which the efficiency of the two mechanisms are the same. The critical number is small when the distance between cells is far, which helps explain why protrusion-based mechanisms are observed in long-range cellular communications.

In many biological systems, natural selection acts simultaneously on multiple levels of organization. This scenario typically presents an evolutionary conflict between the incentive of individuals to cheat and the collective incentive to establish cooperation within a group. Generalizing previous work on multilevel selection in evolutionary game theory, we consider a hyperbolic PDE model of a group-structured population, in which members within a single group compete with each other for individual-level replication; while the group also competes against other groups for group-level replication. We derive a threshold level of the relative strength of between-group competition such that defectors take over the population below the threshold while cooperation persists in the long-time population above the threshold. Under stronger assumptions on the initial distribution of group compositions, we further prove that the population converges to a steady state density supporting cooperation for between-group selection strength above the threshold. We further establish long-time bounds on the time-average of the collective payoff of the population, showing that the long-run population cannot outperform the payoff of a full-cooperator group even in the limit of infinitely-strong between-group competition. When the group replication rate is maximized by an intermediate level of within-group cooperation, individual-level selection casts a long shadow on the dynamics of multilevel selection: no level of between-group competition can erase the effects of the individual incentive to defect. We further extend our model to study the case of multiple types of groups, showing how the games that groups play can coevolve with the level of cooperation.

A selfish learner seeks to maximize their own success, disregarding others. When success is measured as payoff in a game played against another learner, mutual selfishness typically fails to produce the optimal outcome for a pair of individuals. However, learners often operate in populations, and each learner may have a limited duration of interaction with any other individual. Here, we compare selfish learning in stable pairs to selfish learning with stochastic encounters in a population. We study gradient-based optimization in repeated games like the prisoner's dilemma, which feature multiple Nash equilibria, many of which are suboptimal. We find that myopic, selfish learning, when distributed in a population via ephemeral encounters, can reverse the dynamics that occur in stable pairs. In particular, when there is flexibility in partner choice, selfish learning in large populations can produce optimal payoffs in repeated social dilemmas. This result holds for the entire population, not just for a small subset of individuals. Furthermore, as the population size grows, the timescale to reach the optimal population payoff remains finite in the number of learning steps per individual. While it is not universally true that interacting with many partners in a population improves outcomes, this form of collective learning achieves optimality for several important classes of social dilemmas. We conclude that na\"{i}ve learning can be surprisingly effective in populations of individuals navigating conflicts of interest., Comment: 33 pages; final version

Significance Cell migration requires energy, but the metabolic cost of migration has not been quantitatively explored in detail. Here, we use a 2-phase model of the cell cytoplasm to compute cell velocities and energy efficiencies during cell movement. This model predicts that actin polymerization-driven migration is very inefficient in high-hydraulic-resistance environments. Instead, cells can adopt the water-driven mechanism. Therefore, the energetics and mechanical efficiencies of cell movement are predicted to depend on the physical environment., In this work, we explore fundamental energy requirements during mammalian cell movement. Starting with the conservation of mass and momentum for the cell cytosol and the actin-network phase, we develop useful identities that compute dissipated energies during extensions of the cell boundary. We analyze 2 complementary mechanisms of cell movement: actin-driven and water-driven. The former mechanism occurs on 2-dimensional cell-culture substrate without appreciable external hydraulic resistance, while the latter mechanism is prominent in confined channels where external hydraulic resistance is high. By considering various forms of energy input and dissipation, we find that the water-driven cell-migration mechanism is inefficient and requires more energy. However, in environments with sufficiently high hydraulic resistance, the efficiency of actin-polymerization-driven cell migration decreases considerably, and the water-based mechanism becomes more efficient. Hence, the most efficient way for cells to move depends on the physical environment. This work can be extended to higher dimensions and has implication for understanding energetics of morphogenesis in early embryonic development and cancer-cell metastasis and provides a physical basis for understanding changing metabolic requirements for cell movement in different conditions.

We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on R n with n ≥ 2 and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow V γ = κ γ + c with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.

All mammalian cells live in the aqueous medium, yet for many cell biologists, water is a passive arena in which proteins are the leading players that carry out essential biological functions. Recent studies, as well as decades of previous work, have accumulated evidence to show that this is not the complete picture. Active fluxes of water and solutes of water can play essential roles during cell shape changes, cell motility and tissue function, and can generate significant mechanical forces. Moreover, the extracellular resistance to water flow, known as the hydraulic resistance, and external hydraulic pressures are important mechanical modulators of cell polarization and motility. For the cell to maintain a consistent chemical environment in the cytoplasm, there must exist an intricate molecular system that actively controls the cell water content as well as the cytoplasmic ionic content. This system is difficult to study and poorly understood, but ramifications of which may impact all aspects of cell biology from growth to metabolism to development. In this Review, we describe how mammalian cells maintain the cytoplasmic water content and how water flows across the cell surface to drive cell movement. The roles of mechanical forces and hydraulic pressure during water movement are explored.

We investigate the motion of a thin rigid body in Stokes flow and the corresponding slender body approximation used to model sedimenting fibers. In particular, we derive a rigorous error bound comparing a regularized version of the rigid slender body approximation to the classical PDE for rigid motion in the case of a closed loop with constant radius. Our main tool is the slender body PDE framework established by Mori et al. (Commun Pure Appl Math, 2019. https://doi.org/10.1002/cpa.21872; Arch Ration Mech Anal 235(3):1905–1978, 2020), which we adapt to the rigid setting.

For a chemical signal to propagate across a cell, it must navigate a tortuous environment involving a variety of organelle barriers. In this work we study mathematical models for a basic chemical signal, the arrival times at the nuclear membrane of proteins that are activated at the cell membrane and diffuse throughout the cytosol. Organelle surfaces within human B cells are reconstructed from soft X-ray tomographic images, and modeled as reflecting barriers to the molecules’ diffusion. We show that signal inactivation sharpens signals, reducing variability in the arrival time at the nuclear membrane. Inactivation can also compensate for an observed slowdown in signal propagation induced by the presence of organelle barriers, leading to arrival times at the nuclear membrane that are comparable to models in which the cytosol is treated as an open, empty region. In the limit of strong signal inactivation this is achieved by filtering out molecules that traverse non-geodesic paths., Author summary The inside of cells is a complex spatial environment, filled with organelles, filaments and proteins. It is an open question how cell signaling pathways function robustly in the presence of such spatial heterogeneity. In this work we study how organelle barriers influence the most basic of chemical signals; the diffusive propagation of an activated protein from the cell membrane to nucleus. Three-dimensional B cell organelle and membrane geometries reconstructed from soft X-ray tomographic images are used in building mathematical models of the signal propagation process. Our models demonstrate that organelle barriers significantly increase the time required for a diffusing protein to traverse from the cell membrane to nucleus when compared to a cell with an empty cytosolic space. We also show that signal inactivation, a fundamental component of all signaling pathways, can provide robustness in the signal arrival time in two ways. Increasing rates of signal inactivation reduce variability in the arrival time, while also dramatically reducing the degree to which organelle barriers increase the arrival time (in comparison to a cell with an empty cytosol).

In this paper, we have investigated hydration structures of the stable and the metastable acetaminophen crystals (Form I and Form II, respectively) by frequency modulation atomic force microscopy w...

Background Recent years have witnessed an upsurge in the development of methods for estimating current source densities (CSDs) in the neocortical tissue from their recorded local field potential (LFP) reflections using microelectrode arrays. Among these, methods utilizing linear arrays work under the assumption that CSDs vary as a function of cortical depth; whereas they are constant in the tangential direction, infinitely or in a confined cylinder. This assumption is violated in the analysis of neuronal activity propagating along the neocortical sheet, e.g. propagation of alpha waves or cortical spreading depression. New method Here, we developed a novel mathematical method (waveCSD) for CSD analysis of LFPs associated with a planar wave of neocortical neuronal activity propagating at a constant velocity towards a linear probe. Results Results show that the algorithm is robust to the presence of noise in LFP data and uncertainties in knowledge of propagation velocity. Also, results show high level of accuracy of the method in a wide range of electrode resolutions. Using in vivo experimental recordings from the rat neocortex, we employed waveCSD to characterize transmembrane currents associated with cortical spreading depressions. Comparison with existing method(s) Simulation results indicate that waveCSD has a significantly higher reconstruction accuracy compared to the widely-used inverse CSD method (iCSD), and the regularized kernel CSD method (kCSD), in the analysis of CSDs originating from propagating neuronal activity. Conclusions The waveCSD method provides a theoretical platform for estimation of transmembrane currents from their LFPs in experimental paradigms involving wave propagation.

We improved the crystal quality of the metastable phase of a pharmaceutical compound acetaminophen (form II) using solution-mediated phase transformation (SMPT) combined with the control of supersaturation and crystal size. Form II crystals grew at low supersaturation via SMPT of the trihydrate at low temperature (10 °C). While crystal growth rate was low compared to the 20 °C condition, crystals had many defects such as solution inclusions or cracks at 20 °C, but few at 10 °C. The ratio of form II crystals with any defects was small (12%), about 1/3 that of the 20 °C condition. Since the concentration gradient on the crystal surface was relatively small at low bulk supersaturation, high-quality crystals grew uniformly. However, large crystals (>∼0.8 mm) were still defective even at low supersaturation, because concentration gradient is large on large surfaces. Using a new seed-dipping process, obtained crystals were small and uniform in size (∼0.4 mm), and the ratio of defective crystals was reduced to 6%. Powder X-ray diffraction analysis revealed that form II crystals were ∼6 times more temporally stable than in the 20 °C condition. We found that high-quality crystals of metastable phase with high temporal stability could be realized by low supersaturation and reduction of the crystal size via SMPT.

Cortical spreading depression (SD) is a spreading disruption of ionic homeostasis in the brain during which neurons experience complete and prolonged depolarizations. SD is the basis of migraine aura and is increasingly associated with many other brain pathologies. Here, we study the role of glutamate and NMDA receptor dynamics in the context of an ionic electrodiffusion model. We perform simulations in one (1D) and two (2D) spatial dimension. Our 1D simulations reproduce the “inverted saddle” shape of the extracellular voltage signal for the first time. Our simulations suggest that SD propagation depends on two overlapping mechanisms; one dependent on extracellular glutamate diffusion and NMDA receptors and the other dependent on extracellular potassium diffusion and persistent sodium channel conductance. In 2D simulations, we study the dynamics of spiral waves. We study the properties of the spiral waves in relation to the planar 1D wave, and also compute the energy expenditure associated with the recurrent SD spirals., Author summary Cortical spreading depression is a wave of neuronal silencing and ion concentration changes that sweeps slowly through the brain. It is the basis of migraine aura, in which a migraine patient sees a defect move through ones visual field 30 minutes to an hour prior to the headache attack. In this paper, we study the mechanisms by which cortical spreading depression travels as a wave through the brain. We construct a detailed mathematical model based on the physics of ion movement as well as what is known about the molecular players of cortical spreading depression. We build a simulation program that successfully solves the resulting set of highly coupled equations. We find that cortical spreading depression can propagate as a wave by distinct but overlapping mechanisms. We also simulate spiral cortical spreading depression waves and study their properties.

Osmotic forces and solute diffusion are increasingly seen as playing a fundamental role in cell movement. Here, we present a numerical method that allows for studying the interplay between diffusive, osmotic and mechanical effects. An osmotically active solute obeys a advection–diffusion equation in a region demarcated by a deformable membrane. The interfacial membrane allows transmembrane water flow which is determined by osmotic and mechanical pressure differences across the membrane. The numerical method is based on an immersed boundary method for fluid–structure interaction and a Cartesian grid embedded boundary method for the solute. We demonstrate our numerical algorithm with the test case of an osmotic engine, a recently proposed mechanism for cell propulsion.

Side-to-side anastomoses are more frequently performed. This method is relatively simple, timesaving, and can be applied in a variety of settings where anastomoses are necessary. Despite some advances, stapled anastomoses have inherent weak points. With any type of side-to-side anastomosis, the crotch is formed at the tip portion of the stapler; this crotch area is the weak point. We describe a novel surgical technique for side-to-side anastomosis using a reinforced crotch. After distal gastrectomy, a small incision was made on the greater curvature of the remnant stomach and the posterior side of the duodenum. We use a 60-mm linear stapler, namely, the Endo GIATM Reinforce Reload (Covidien Japan). Therefore, the stapler length inserted into the stomach and duodenum is 45 mm. The reinforce reloads have a preloaded reinforcement material on the both anvil and cartridge. One or two supporting sutures were added to the center of the common enterotomy. Pulling up on the two pieces of Neoveil® and the sutures, the common enterotomy was closed with a linear stapler. Delta-shaped anastomosis is then accomplished. We also use a reinforced reload in a side-to-side anastomosis between two segments of jejunum. Neoveil® is thin and soft; we found no problem using either a hand thrown suture or stapler to close the entry hole. In fact, we found that the excess Neoveil® is very useful. Neoveil® can be a substitute for the sutures. Only one or two hand thrown sutures were then necessary to close the common enterotomy. And we do not add the reinforce suture of the crotch at all. This novel method is a simple and useful method and can be used in side-to-side anastomoses in a variety of settings.

Highlights • The indications for stripping of the esophagus have decreased due to the widespread of endoscopic mucosal resection and thoracoscopic surgery. • Even if indications for this procedure have decreased, this is an important option in the armamentarium of the esophageal surgeon., Introduction Patients with esophageal cancer frequently cannot tolerate thoracotomy due to their overall debilitated condition. Moreover, some patients have severe adhesions in the thoracic cavity. Eversion stripping of the esophagus is an option for resection in these patients. Presentation of case A 64-year-old man was admitted to our institution with the chief complaint of epigastric pain. Endoscopic examination showed a protruding lesion 22 cm from the incisors, with a superficial and circumferential mucosal irregularity on the distal side of the lesion. Biopsy revealed squamous cell carcinoma. Clinical stage was T1b(sm)N0M0, cStage I. In addition to the poor pulmonary status of the patient, adhesions in the intrathoracic cavity were predicted. The decision was made to perform esophageal resection without a thoracotomy. In order to ensure complete invagination of the esophagus, the esophagus was insufflated prior to stripping. The stripping process was observed with a gastroscope. During the stripping, the esophagus did not bunch up, and stripping was smooth and with minimal resistance. Discussion The stripping resection of the esophagus is an important option for the esophageal surgeon. In this case report, we describe a new eversion stripping method of the esophagus. This easy and reliable stripping method incorporates intraesophageal insufflation. Conclusion The indications for blunt esophageal dissection without thoracotomy have been decreasing. On the other hand, our method seems to be useful in optimal case of stripping of esophagus.

When a poroelastic spherical drop is under a uniaxial extension flow or a shear flow, the elastic network deforms in response to the external fluid viscous stress. A small-deformation analysis is conducted to investigate effects of interfacial slip and permeability on flow around the drop.

Slender body theory is a commonly used approximation in computational models of thin fibers in viscous fluids, especially in simulating the motion of cilia or flagella in swimming microorganisms. In Mori et al. (Commun Pure Appl Math, 2018. arXiv:1807.00178), we developed a PDE framework for analyzing the error introduced by the slender body approximation for closed-loop fibers with constant radius $$\varepsilon $$, and showed that the difference between our closed-loop PDE solution and the slender body approximation is bounded by an expression proportional to $$\varepsilon |\log \varepsilon |$$. Here we extend the slender body PDE framework to the free endpoint setting, which is more physically relevant from a modeling standpoint but more technically demanding than the closed loop analysis. The main new difficulties arising in the free endpoint setting are defining the endpoint geometry, identifying the extent of the 1D slender body force density, and determining how the well-posedness constants depend on the non-constant fiber radius. Given a slender fiber satisfying certain geometric constraints at the filament endpoints and a one-dimensional force density satisfying an endpoint decay condition, we show a bound for the difference between the solution to the slender body PDE and the slender body approximation in the free endpoint setting. The bound is a sum of the same $$\varepsilon |\log \varepsilon |$$ term appearing in the closed loop setting and an endpoint term proportional to $$\varepsilon $$, where $$\varepsilon $$ is now the maximum fiber radius.

Cortical spreading depression (SD) is a spreading disruption in brain ionic homeostasis during which neurons experience complete and prolonged depolarizations. SD is generally believed to be the physiological substrate of migraine aura and is associated with many other brain pathologies. Here, we perform simulations with a model of SD treating brain tissue as a triphasic continuum of neurons, glia and the extracellular space. A thermodynamically consistent incorporation of the major biophysical effects, including ionic electrodiffusion and osmotic water flow, allows for the computation of important physiological variables including the extracellular voltage (DC) shift. A systematic parameter study reveals that glia can act as both a disperser and buffer of potassium in SD propagation. Furthermore, we show that the timing of the DC shift with respect to extracellular $$\hbox {K}^{+}$$ rise is highly dependent on glial parameters, a result with implications for the identification of the propagating mechanism of SD.

Decreased and heterogeneous expression of connexin 43 (Cx43) are common features in animal heart failure models. Ephpatic coupling, which relies on the presence of junctional cleft space between the ends of adjacent cells, has been suggested to play a more active role in mediating intercellular electrical communication when gap junctions are reduced. To better understand the interplay of Cx43 expression and ephaptic coupling on cardiac conduction during heart failure, we performed numerical simulations on our model when Cx43 expression is reduced and heterogeneous. Under severely reduced Cx43 expression, we identified three new phenomena in the presence of ephaptic coupling: alternating conduction, in which ephaptic and gap junction-mediated mechanisms alternate; instability of planar fronts; and small amplitude action potential (SAP), which has a smaller potential amplitude than the normal action potential. In the presence of heterogeneous Cx43 expression, ephaptic coupling can either prevent or promote conduction block (CB) depending on the Cx43 knockout (Cx43KO) content. When Cx43KO content is relatively high, ephaptic coupling reduces the probabilities of CB. However, ephaptic coupling promotes CB when Cx43KO and wild type cells are mixed in roughly equal proportion, which can be attributed to an increase in current-to-load mismatch.

Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three-dimensional fiber, in part due to the fact that simply prescribing data along a one-dimensional curve does not result in a well-posed boundary value problem for the Stokes equations in $\mathbb{R}^3$. Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well-posedness framework, allow us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry.

The electroneutral model describes cellular electrical activity, accounting for ionic concentration dynamics without resolution of the fine spatial scales of the space-charge layer. This is done by asserting that the ionic solution is electrically neutral at each point in space. However, electroneutrality is inconsistent with the original boundary conditions at cell membranes. We consider three separate methods of resolving this inconsistency that result in well-posed models that are accurate approximations to a detailed model in which the space-charge layer is fully resolved. A particular electrodiffusion problem is utilized to make the discussion specific. © 2015 Wiley Periodicals, Inc.

Ionic electrodiffusion and osmotic water flow are central processes in many physiological systems. We formulate a system of partial differential equations that governs ion movement and water flow in biological tissue. A salient feature of this model is that it satisfies a free energy identity, ensuring the thermodynamic consistency of the model. A numerical scheme is developed for the model in one spatial dimension and is applied to a model of cortical spreading depression, a propagating breakdown of ionic and cell volume homeostasis in the brain., submitted for publication, Aug. 28, 2014

The Taylor–Melcher (TM) model is the standard model for describing the dynamics of poorly conducting leaky dielectric fluids under an electric field. The TM model treats the fluids as ohmic conductors, without modelling the underlying ion dynamics. On the other hand, electrodiffusion models, which have been successful in describing electrokinetic phenomena, incorporate ionic concentration dynamics. Mathematical reconciliation of the electrodiffusion picture and the TM model has been a major issue for electrohydrodynamic theory. Here, we derive the TM model from an electrodiffusion model in which we explicitly model the electrochemistry of ion dissociation. We introduce salt dissociation reaction terms in the bulk electrodiffusion equations and take the limit in which the salt dissociation is weak; the assumption of weak dissociation corresponds to the fact that the TM model describes poor conductors. Together with the assumption that the Debye length is small, we derive the TM model with or without the surface charge convection term depending upon the scaling of relevant dimensionless parameters. An important quantity that emerges is the Galvani potential (GP), the jump in voltage across the liquid–liquid interface between the two leaky dielectric media; the GP arises as a natural consequence of the interfacial boundary conditions for the ionic concentrations, and is absent under certain parametric conditions. When the GP is absent, we recover the TM model. Our analysis also reveals the structure of the Debye layer at the liquid–liquid interface, which suggests how interfacial singularities may arise under strong imposed electric fields. In the presence of a non-zero GP, our model predicts that the liquid droplet will drift under an imposed electric field, the velocity of which is computed explicitly to leading order.

With advances of combined modality therapy, prognoses in esophageal cancer have been improving. After resection of esophageal cancer, the development of gastric tube cancer is a risk. While such cancer in an early stage can be cured endoscopically, total gastric tube resection is indicated in advanced stages. A 68-year-old man underwent subtotal esophagectomy reconstructed with a gastric tube through the retrosternal route. Gastric cancer was found one and a half years postoperatively. The gastric tube was resected without sternotomy. This is the first report of a patient undergoing resection of the gastric tube reconstructed through the retrosternal route without sternotomy.

We analyse a model of polyelectrolyte gels that was proposed by the authors in previous work. We first demonstrate that the model can be derived using Onsager's variational principle, a general procedure for obtaining equations in soft condensed matter physics. The model is shown to have a unique steady state under the assumption that a suitably defined mechanical energy density satisfies a convexity condition. We then perform a detailed study of the stability of the steady state in the spatially one-dimensional case, obtaining bounds on the relaxation rate. Numerical simulations for the spatially one-dimensional problem are presented, confirming the analytical calculations on stability.

In this paper, we analyze the convergence of the immersed boundary method as applied to a static Stokes flow problem. Using estimates obtained in [Y. Liu and Y. Mori, SIAM J. Numer. Anal., 50 (2012), pp. 2986--3015], we consider a problem in which a $d$-dimensional structure is immersed in an $n$-dimensional domain, and prove error estimates for both the pressure and the velocity field in the $L^p(1\leq p\leq \infty)$ norm. One interesting consequence of our analysis is that the asymptotic error rates in the $L^1$ norm do not depend on either $d$ or $n$ and in the $L^p$ ($p>1$) norm they only depend on $n-d$. The resulting estimates are checked numerically for optimality.

We analyze the stability and convergence of first-order accurate and second-order accurate timestepping schemes for the Navier--Stokes equations with variable viscosity. These schemes are characterized by a mixed implicit/explicit treatment of the viscous term, in which a numerical parameter, $\lambda$, determines the degree of splitting between the implicit and explicit contributions. The reason for this splitting is that it avoids the need to solve computationally expensive linear systems that may change at each timestep. Provided the parameter $\lambda$ is within a permissible range, we prove that the first-order accurate and second-order accurate schemes are convergent. We show further that the efficiency of the second-order accurate scheme depends on how $\lambda$ is chosen within the permissible range, and we discuss choices that work well in practice. We use parameters motivated by this analysis to simulate internal gravity waves, which arise in stratified fluids with variable density. We examine h...

We derive a model of the coupled mechanical and electrochemical effects of polyelectrolyte gels. We assume that the gel, which is immersed in a fluid domain, is an immiscible and incompressible mixture of a solid polymeric component and the fluid. As the gel swells and deswells, the gel-fluid interface can move. Our model consists of a system of partial differential equations for mass and linear momentum balance of the polymer and fluid components of the gel, the Navier--Stokes equations in the surrounding fluid domain, and the Poisson--Nernst--Planck equations for the ionic concentrations on the whole domain. These are supplemented by a novel and general class of boundary conditions expressing mass and linear momentum balance across the moving gel-fluid interface. Our boundary conditions include the permeability boundary conditions proposed in earlier studies. A salient feature of our model is that it satisfies a free energy dissipation identity, in accordance with the second law of thermodynamics. We al...

We present a general variable viscosity and variable density immersed boundary method that is first-order accurate in the variable density case and, for problems possessing sufficient regularity, second-order accurate in the constant density case. The viscosity and density are considered material properties and are defined by a dynamically updated tesselation. Empirical convergence rates are reported for a test problem of a two-dimensional viscoelastic shell with spatially varying material properties. The reduction to first-order accuracy in the variable density case can be avoided by using an iterative scheme, although this approach may not be efficient enough for practical use. In our time-stepping scheme, both the inertial and viscous terms are split into two parts: a constant-coefficient part that is treated implicitly, and a variable-coefficient part that is treated explicitly. This splitting allows the resulting equations to be solved efficiently using fast constant-coefficient linear solvers, and i...

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.