Search

We analyze a class of cell-bulk coupled PDE-ODE models, motivated by quorum-sensing and diffusion-mediated behavior in microbial systems, that characterize communication between localized spatially...

Localized spot patterns, where one or more solution components concentrate at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in...

Spatially localized 2-D spot patterns occur for a wide variety of two component reaction-diffusion systems in the singular limit of a large diffusivity ratio. Such localized, far-from-equilibrium, patterns are known to exhibit a wide range of different instabilities such as breathing oscillations, spot annihilation, and spot self-replication behavior. Prior numerical simulations of the Schnakenberg and Brusselator systems have suggested that a localized peanut-shaped linear instability of a localized spot is the mechanism initiating a fully nonlinear spot self-replication event. From a development and implementation of a weakly nonlinear theory for shape deformations of a localized spot, it is shown through a normal form amplitude equation that a peanut-shaped linear instability of a steady-state spot solution is always subcritical for both the Schnakenberg and Brusselator reaction-diffusion systems. The weakly nonlinear theory is validated by using the global bifurcation software {\em pde2path} [H.~Uecker et al., Numerical Mathematics: Theory, Methods and Applications, {\bf 7}(1), (2014)] to numerically compute an unstable, non-radially symmetric, steady-state spot solution branch that originates from a symmetry-breaking bifurcation point.

For three specific singularly perturbed two-component reaction diffusion systems in a bounded two-dimensional domain admitting localized multispot patterns, we provide a detailed analysis of the pa...

We analyze the existence and linear stability of steady-state localized hotspot patterns for a 1-D three-component singularly perturbed reaction-diffusion (RD) system modeling urban crime in the pr...

A linear stability analysis of localized spike solutions to the singularly perturbed two-component Gierer--Meinhardt (GM) reaction-diffusion (RD) system with a fixed time delay $T$ in the nonlinear reaction kinetics is performed. Our analysis of this model is motivated by the computational study of Lee, Gaffney, and Monk [Bull. Math. Bio., 72 (2010), pp. 2139--2160] on the effect of gene expression time delays on spatial patterning for both the GM model and some related RD models. It is shown that the linear stability properties of such localized spike solutions are characterized by the discrete spectra of certain nonlocal eigenvalue problems (NLEP). Phase diagrams consisting of regions in parameter space where the steady-state spike solution is linearly stable are determined for various limiting forms of the GM model in both 1-dimensional and 2-dimensional domains. On the boundary of the region of stability, the spike solution is found to undergo a Hopf bifurcation. For a special range of exponents in the nonlinearities for the 1-dimensional GM model, and assuming that the time delay only occurs in the inhibitor kinetics, this Hopf bifurcation boundary is readily determined analytically. For this special range of exponents, the challenging problem of locating the discrete spectrum of the NLEP is reduced to the much simpler problem of locating the roots to a simple transcendental equation in the eigenvalue parameter. By using a hybrid analytical-numerical method, based on a parametrization of the NLEP, it is shown that qualitatively similar phase diagrams occur for general GM exponent sets and for the more biologically relevant case where the time delay occurs in both the activator and inhibitor kinetics. Overall, our results show that there is a critical value $T_{\star}$ of the delay for which the spike solution is unconditionally unstable for $T>T_{*}$, and that the parameter region where linear stability is assured is, in general, rather limited. A comparison of the theory with full numerical results computed from the RD system with delayed reaction kinetics for a particular parameter set suggests that the Hopf bifurcation can be subcritical, leading to a global breakdown of a robust spatial patterning mechanism.

On a bounded three-dimensional domain $\Omega$, a hybrid asymptotic-numerical method is employed to analyze the existence, linear stability, and slow dynamics of localized quasi-equilibrium multispot patterns of the Schnakenberg activator-inhibitor model with bulk feed-rate $A$ in the singularly perturbed limit of small diffusivity $\varepsilon^2$ of the activator component. By approximating each spot as a Coulomb singularity, a nonlinear system of equations is formulated for the strength of each spot. To leading order in $\varepsilon$, two types of solutions are identified: symmetric patterns for which all strengths are identical, and asymmetric patterns for which each strength takes on one of two distinct values. The $\mathcal{O}(\varepsilon)$ correction to the strengths is found to depend on the spatial configuration of the spots through a certain Neumann Green's matrix $\mathcal{G}$. When $\mathbf{e} = (1,\dots,1)^T$ is not an eigenvector of $\mathcal{G}$, a detailed numerical and (in the case of two ...

We analyze a singularly perturbed reaction-diffusion system in the semistrong diffusion regime in two spatial dimensions where an activator species is localized to a closed curve, while the inhibitor species exhibits long-range behavior over the domain. In the limit of small activator diffusivity, we derive a new moving boundary problem characterizing the slow time evolution of the curve, which is defined in terms of a quasi-steady state inhibitor diffusion field and its properties on the curve. Numerical results from this curve evolution problem are illustrated for the Gierer--Meinhardt model with saturation (GMS) in the activator kinetics. A detailed analysis of the existence, stability, and dynamics of ring and near-ring solutions for the GMS model is given, whereby the activator concentrates on a thin ring concentric within a circular domain. A key new result for this ring geometry is that by including activator saturation there is a qualitative change in the phase portrait of ring equilibria, in that...

We study the first passage time problem for a diffusing molecule in an enclosed region to hit a small spherical target whose surface contains many small absorbing traps. This study is motivated by two examples of cellular transport. The first is the intracellular process through which proteins transit from the cytosol to the interior of the nucleus through nuclear pore complexes that are distributed on the nuclear surface. The second is the problem of chemoreception, in which cells sense their surroundings through diffusive contact with receptors distributed on the cell exterior. Using a matched asymptotic analysis in terms of small absorbing pore radius, we derive and numerically verify a high order expansion for the capacitance of the structured target which incorporates surface effects and gives explicit information on interpore interaction through a Coulomb-type discrete energy with additional logarithmic dependencies. In the large $N$ dilute surface trap fraction limit, a single homogenized Robin boundary condition $ \partial_n v + \kappa v = 0$ is derived in which $\kappa$ depends on the total absorbing fraction, the characteristic pore scale, and parameters relating to interpore interactions.

Oscillatory dynamics associated with the coupled membrane-bulk PDE-ODE model of Gomez-Marin, Garcia-Ojalvon, and Sancho [Phys. Rev. Lett., 98 (2007), 168303] in one spatial dimension is analyzed using a combination of asymptotic analysis, linear stability theory, and numerical bifurcation software. The mathematical model consists of two dynamically active membranes with Fitzhugh--Nagumo kinetics, separated spatially by a distance $L$, that are coupled together through a diffusion field that occupies the bulk region $0

The mixed-order eigenvalue problem $-\delta \Delta^2 u + \Delta u + \lambda u = 0$ with $\delta>0$, modeling small amplitude vibrations of a thin plate, is analyzed in a bounded two-dimensional domain $\Omega$ that contains a single small hole of radius $\varepsilon$ centered at some $x_0\in \Omega$. Clamped conditions are imposed on the boundary of $\Omega$ and on the boundary of the small hole. In the limit $\varepsilon\to 0$, and for $\delta={\mathcal O}(1)$, the limiting problem for $u$ must satisfy the additional point constraint $u(x_0)=0$. To determine how the eigenvalues of the Laplacian in a domain with a small hole are perturbed by adding the small fourth-order term $-\delta \Delta^2 u$, together with an additional boundary condition on $\partial\Omega$ and on the hole boundary, the asymptotic behavior of the eigenvalues of the mixed-order eigenvalue problem are studied in the dual limit $\varepsilon\to 0$ and $\delta\to 0$. Leading-order behaviors of eigenvalues are determined for three ranges ...

Calculating the time required for a diffusing object to reach a small target within a larger domain is a feature of a large class of modeling and simulation efforts in biology. Here, we are motivated by the motion of a T cell of the immune system seeking a particular antigen-presenting cell within a large lymph node. The precise nature of the cell motion at the outer boundary of the lymph node is not completely understood in terms of how cells choose to remain within a given lymph node, or exit. In previous work, we and others have studied diffusive motion to a small trap. We extend this previous work to analyze models where the diffusing object may exit the outer boundary of the domain (in this case, the lymph node). This is modeled by a Robin boundary condition on the surface of the lymph node. For the general problem of small traps inside a three-dimensional domain that has a partially sticky or absorbent domain boundary, the method of matched asymptotic expansions is used to calculate the mean and variance of the conditional first passage time for the T cell to reach a specific target trap. Our results are illustrated explicitly for the idealized situation of a spherical lymph node containing small spherically shaped traps, and are verified for a radially symmetric geometry with one trap at the origin where exact solutions are available. Mathematically, our analysis extends previous work on the calculation of the mean first passage time by allowing for a sticky boundary and by calculating conditional statistics of the diffusion process. Finally, our results are interpreted and applied to the context of T cell biology.

Spatial-temporal dynamics associated with a class of coupled membrane-bulk PDE-ODE models in one spatial dimension is analyzed using a combination of linear stability theory, numerical bifurcation software, and full time-dependent simulations. In our simplified one-dimensional setting, the mathematical model consists of two dynamically active membranes, separated spatially by a distance $2L$, that are coupled together through a linear bulk diffusion field, with a fixed bulk decay rate. The coupling of the bulk and active membranes arises through both nonlinear flux boundary conditions for the bulk diffusion field and from feedback terms, depending on the local bulk concentration, to the dynamics on each membrane. For this class of models, it is shown both analytically and numerically that bulk diffusion can trigger a synchronous oscillatory instability in the temporal dynamics associated with the two active membranes. For the case of a single active component on each membrane, and in the limit $L\to \inft...

In the singularly perturbed limit of an asymptotically small diffusivity ratio ${\varepsilon}^2$, the existence and stability of localized quasi-equilibrium multispot patterns is analyzed for the Brusselator reaction-diffusion model on the unit sphere. Formal asymptotic methods are used to derive a nonlinear algebraic system that characterizes quasi-equilibrium spot patterns and to formulate eigenvalue problems governing the stability of spot patterns to three types of “fast” ${\mathcal O}(1)$ time-scale instabilities: self-replication, competition, and oscillatory instabilities of the spot amplitudes. The nonlinear algebraic system and the spectral problems are then studied using simple numerical methods, with emphasis on the special case where the spots have a common amplitude. Overall, the theoretical framework provides a hybrid asymptotic-numerical characterization of the existence and stability of spot patterns that is asymptotically correct to within all logarithmic correction terms in powers of $\n...

A mathematical analysis is undertaken of a Schnakenberg reaction-diffusion system in one dimension with a spatial gradient governing the active reaction. This system has previously been proposed as a model of the initiation of hairs from the root epidermis Arabidopsis, a key cellular-level morphogenesis problem. This process involves the dynamics of the small G-proteins, Rhos of plants, which bind to form a single localized patch on the cell membrane, prompting cell wall softening and subsequent hair growth. A numerical bifurcation analysis is presented as two key parameters, involving the cell length and the overall concentration of the auxin catalyst, are varied. The results show hysteretic transitions from a boundary patch to a single interior patch, and to multiple patches whose locations are carefully controlled by the auxin gradient. The results are confirmed by an asymptotic analysis using semistrong interaction theory, leading to closed form expressions for the patch locations and intensities. A c...

The mean first passage time (MFPT) is calculated for a Brownian particle in a bounded two-dimensional domain that contains N small nonoverlapping absorbing windows on its boundary. The reciprocal o...

Intracellular signaling gradients naturally arise through a local activation of a dif- fusible signaling molecule, e.g., by a localized kinase, and a subsequent deactivation at a distant cellular site, e.g., by a cytosolic phosphatase. Here, we consider a spherical cell containing a finite number of small spherical compartments where a signaling molecule becomes activated by a localized enzyme. For the activation rate two cases are considered: a saturated enzyme with a constant rate and an unsaturated enzyme with a linear rate. Using the method of matched asymptotic expansions, we derive approximate solutions of the steady-state diffusion equation with a linear deactivation rate to obtain the three-dimensional concentration profile of activated signaling molecules inside the cell. Depending on the diffusion length of the signaling molecule, the profiles decay either exponentially or algebraically, where the mode of decay is described by an associated Green's function. Our analysis provides simple expressions for the local concentration profile in the neighborhood of a signaling compartment, which can be used to estimate the amplitude and the extent of the gradients in the respective regimes. The global concentration profiles also depend on the cell size and the particular spatial arrangement of the compartments relative to each other and to the cell boundary.

A common scenario in cellular signal transduction is that a diffusing surface-bound molecule must arrive at a localized signaling region on the cell membrane before the signaling cascade can be completed. The question then arises of how quickly such signaling molecules can arrive at newly formed signaling regions. Here, we attack this problem by calculating asymptotic results for the mean first passage time for a diffusing particle confined to the surface of a sphere, in the presence of N partially absorbing traps of small radii. The rate at which the small diffusing molecule becomes captured by one of the traps is determined by asymptotically calculating the principal eigenvalue for the Laplace operator on the sphere with small localized traps. The asymptotic analysis relies on the method of matched asymptotic expansions, together with detailed properties of the Green's function for the Laplacian and the Helmholtz operators on the surface of the unit sphere. The asymptotic results compare favorably with ...

We present a model of protein receptor trafficking within the membrane of a cylin- drical dendrite containing small protrusions called spines. Spines are the locus of most excitatory synapses in the central nervous system and act as localized traps for receptors diffusing within the dendritic membrane. We treat the transverse intersection of a spine and dendrite as a spatially extended, partially absorbing boundary and use singular perturbation theory to analyze the steady- state distribution of receptors. We compare the singular perturbation solutions with numerical solutions of the full model and with solutions of a reduced one-dimensional model and find good agreement between them all. We also derive a system of Fokker-Planck equations from our model and use it to exactly solve a mean first passage time (MFPT) problem for a single receptor traveling a fixed axial distance along the dendrite. This is then used to calculate an effective diffusion coefficient for receptors when spines are uniformly distributed along the length of the cable and to show how a nonuniform distribution of spines gives rise to anomalous subdiffusion.

The stability of two different types of stripe solutions that occur for two different forms of the Gierer--Meinhardt (GM) activator-inhibitor model is analyzed in a rectangular domain. For the basic GM model with exponent set $(p,q,r,s)$, representing the powers of certain nonlinear terms in the reaction kinetics, a homoclinic stripe is constructed whereby the activator concentration localizes along the midline of the rectangular domain. In the semistrong regime, characterized by a global variation of the inhibitor concentration across the domain, instability bands with respect to transverse zigzag instabilities and spot-generating breakup instabilities of the homoclinic stripe are determined analytically. In the weak interaction regime, where both the inhibitor and activator concentrations are localized, the spectrum of the linearization of the homoclinic stripe is studied numerically with respect to both breakup and zigzag instabilities. For certain exponent sets near the existence threshold of this hom...

The dynamics and instability mechanisms of both one- and two-spike solutions to the Gierer--Meinhardt (GM) and Gray--Scott (GS) models are analyzed on a bounded one-dimensional spatial domain. For each of these nonvariational two-component systems, the semistrong spike-interaction limit where the ratio $O(\varepsilon^{-2})$ of the two diffusion coefficients is asymptotically large is studied. In this limit, differential equations for the spike locations, with speed $O(\varepsilon^2) \ll 1$, are derived. To determine the stability of the spike patterns, nonlocal eigenvalue problems, which depend on the instantaneous spike locations, are derived and analyzed. For these nonlocal eigenvalue problems, it is shown that eigenvalues can enter into the unstable right half-plane either along the real axis or through a Hopf bifurcation, leading to either a competition instability or an oscillatory instability of the spike pattern, respectively. Competition instabilities occur only for two-spike patterns and numerica...

In the limit of small chemoattractant diffusivity $\epsilon$, the existence, stability, and dynamics of spiky patterns in a chemotaxis model are studied in a bounded multidimensional domain. In this model, the transition probability density function $\Phi(w)$ is assumed to have a power law form $\Phi(w)=w^p$, and the production of chemoattractant w is assumed to saturate according to a Michaelis--Menten kinetic function. In the limit $\epsilon \to 0$, it is proved that there is a steady-state single boundary spike solution located at the maximum of the mean curvature of the boundary. Moreover, a steady-state interior spike solution is proved to concentrate at a maximum of the distance function. The single interior spike solution is shown to be metastable for certain ranges of p and the dimension N. The stability of a single boundary spike solution is also analyzed in detail. Finally, a formal asymptotic analysis is used to characterize the metastable interior spike dynamics in both a one-dimensional and a...

A mathematical model of the transport of oxygen from capillaries to skeletal muscle tissue is a diffusion problem in a two-dimensional, bounded domain with Neumann and mixed boundary conditions. We consider N capillaries of small but arbitrary cross-sectional shape and demonstrate, for N > 1, that this is a singular perturbation problem that involves an infinite expansion of logarithmic terms of the small parameter $\varepsilon$, which characterizes the size of the capillary cross sections. For $\varepsilon \ll 1$, we use a hybrid asymptotic-numerical method to calculate the steady-state oxygen partial pressure in the tissue correct to within all logarithmic terms. Our results from this hybrid method illustrate the effect of tissue heterogeneities such as mitochondria, variable permeability of the capillary walls, and the facilitation of oxygen transport by the presence of myoglobin. The results from the hybrid method compare well with full numerical solutions.

An asymptotic reduction of the Gierer--Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity leads to a singularly perturbed nonlocal reaction diffusion equation for the activator concentration. In the limit of small activator diffusivity, a one-spike solution to this nonlocal model is constructed. The spectrum of the eigenvalue problem associated with the linearization of the nonlocal model around such an isolated spike solution is studied in both a one-dimensional and a multidimensional context. It is shown that the principal eigenvalues in the spectrum are exponentially small in the limit of small activator diffusivity. The nonlocal term in the eigenvalue problem is essential for ensuring the existence of such exponentially small principal eigenvalues. These eigenvalues are responsible for the occurrence of an exponentially slow, or metastable, spike-layer motion for the time-dependent problem. Explicit metastable spike dynamics are derived by using a projection method, which...

In a multidimensional domain, the slow motion behavior of internal layer solutions with spherical interfaces, referred to as bubble solutions, is analyzed for the nonlocal Allen–Cahn equation with mass conservation. This problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint. The bubble is shown to drift exponentially slowly across the domain, without change of shape, toward the closest point on the boundary of the domain. An explicit ordinary differential equation for the motion of the center of the bubble is derived by extending, to a multidimensional setting, the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension. An asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point. An analogy between slow bubble motion and the classical exit prob...

The classical problem of slow, steady, two-dimensional flow of a viscous incompressible fluid around an infinitely long straight cylinder is considered. The cylinder cross section is symmetric about the direction of the oncoming stream but otherwise is arbitrary. For low Reynolds number, the well-known singular perturbation analysis for this problem shows that the asymptotic expansions of the drag coefficient and of the flow field start with infinite logarithmic series. We show that the entire infinite logarithmic expansions of the flow field and of the drag coefficient are contained in the solution to a certain related problem that does not involve the cross-sectional shape of the cylinder. The solution to this related problem is computed numerically using a straightforward finite-difference scheme. The drag coefficient for a cylinder of a specific cross-sectional shape, which is asymptotically correct to within all logarithmic terms, is given in terms of a single shape-dependent constant that is determi...

On a semi-infinite domain, an analytical characterization of exponentially slow internal layer motion for the Allen–Cahn equation and for a singularly perturbed viscous shock problem is given. The results extend some previous results that were restricted to a finite geometry. For these slow motion problems, we show that the slow dynamics associated with the semi-infinite domain are not preserved, even qualitatively, by imposing a commonly used form of artificial boundary condition to truncate the semi-infinite domain to a finite domain. This extreme sensitivity to boundary conditions and domain truncation is a direct result of the exponential ill-conditioning of the underlying linearized problem. For Burgers equation, many of the analytical results are verified by calculating certain explicit solutions. Some related ill-conditioned internal layer problems are examined.

This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded o...

The drift-diffusion equations of semiconductor physics, modeling the behavior of buried-channel field-effect transistors, are analyzed using formal perturbation techniques. For large aspect ratio devices, the potential distribution is essentially one-dimensional under the gate and has a boundary layer structure near the source and drain. By extending the results of Ward, Odeh, and Cohen [SIAM J. Appl. Math., 4 (1990), pp. 1099–1125], where a different class of devices was treated, the potential under the gate is resolved in the limit of large doping densities for various values of the gate voltage and implant depth. Using the asymptotic potential, the mobile charge, which is needed for the derivation of the long-channel current-voltage relations, is found using standard techniques in the asymptotic evaluation of integrals. The results of Ward, Odeh, and Cohen are also extended to analyze the potential distribution in the fully two-dimensional regions near the source and drain in equilibrium. In the limit ...