The role of feedback in the formation of morphogen territories
David Iron Adeela Syed Heidi Theisen Tamas Lukacsovich Mehrangiz Naghibi Lawrence J. Marsh Frederic Y. M. Wan Qing Nie
In this paper, we consider a mathematical model for the forma- tion of spatial morphogen territories of two key morphogens: Wingless (Wg) and Decapentaplegic (DPP), involved in leg development of Drosophila. We define a gene regulatory network (GRN) that utilizes autoactivation and cross- inhibition (modeled by Hill equations) to establish and maintain stable bound- aries of gene expression. By computational analysis we find that in the presence of a general activator, neither autoactivation, nor cross-inhibition alone are suf- ficient to maintain stable sharp boundaries of morphogen production in the leg disc. The minimal requirements for a self-organizing system are a coupled system of two morphogens in which the autoactivation and cross-inhibition have Hill coefficients strictly greater than one. In addition, the GRN modeled here describes the regenerative responses to genetic manipulations of positional identity in the leg disc.
keywords: gene regulatory network pattern formation partial differential equations. dynamical systems
Feedback regulation in multistage cell lineages
Wing-Cheong Lo Ching-Shan Chou Kimberly K. Gokoffski Frederic Y.-M. Wan Arthur D. Lander Anne L. Calof Qing Nie
Studies of developing and self-renewing tissues have shown that differentiated cell types are typically specified through the actions of multistage cell lineages. Such lineages commonly include a stem cell and multiple progenitor (transit amplifying; TA) cell stages, which ultimately give rise to terminally differentiated (TD) cells. In several cases, self-renewal and differentiation of stem and progenitor cells within such lineages have been shown to be under feedback regulation. Together, the existence of multiple cell stages within a lineage and complex feedback regulation are thought to confer upon a tissue the ability to autoregulate development and regeneration, in terms of both cell number (total tissue volume) and cell identity (the proportions of different cell types, especially TD cells, within the tissue). In this paper, we model neurogenesis in the olfactory epithelium (OE) of the mouse, a system in which the lineage stages and mediators of feedback regulation that govern the generation of terminally differentiated olfactory receptor neurons (ORNs) have been the subject of much experimental work. Here we report on the existence and uniqueness of steady states in this system, as well as local and global stability of these steady states. In particular, we identify parameter conditions for the stability of the system when negative feedback loops are represented either as Hill functions, or in more general terms. Our results suggest that two factors -- autoregulation of the proliferation of transit amplifying (TA) progenitor cells, and a low death rate of TD cells -- enhance the stability of this system.
keywords: cell lineage olfactory epithelium neurogenesis terminally differentiated cell feedback transit amplifying cell modeling stability neuronal progenitor stem cell
Xiaoying Han Qing Nie
Stochasticity, sometimes referred to as noise, is unavoidable in biological systems. Noise, which exists at all biological scales ranging from gene expressions to ecosystems, can be detrimental or sometimes beneficial by performing unexpected tasks to improve biological functions. Often, the complexity of biological systems is a consequence of dealing with uncertainty and noise, and thus, consideration of noise is necessary in mathematical models. Recent advancement of technology allows experimental measurement on stochastic effects, showing multifaceted and perplexed roles of noise. As interrogating internal or external noise becomes possible experimentally, new models and mathematical theory are needed. Over the past few decades, stochastic analysis and the theory of nonautonomous and random dynamical systems have started to show their strong promise and relevance in studying complex biological systems. This special issue represents a collection of recent advances in this emerging research area.

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Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory
Christopher Rackauckas Qing Nie

Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.

keywords: Stochastic differential equations adaptive methods rejection sampling embedded algorithms stochastic Runge-Kutta strong approximation
Qing Nie Ka Kit Tung
This special issue of the Discrete and Continuous Dynamical Systems is dedicated to Professor Frederic Yui-Ming Wan on the occasion of his seventieth birthday. For this special occasion, a conference on Biology and Mechanics: Applications of Mathematics and Computations was held on May 25 and 26, 2006 at the Beckman Center for National Academies at Irvine. This volume is a compilation of contributions from friends, colleagues, and collaborators of Fred Wan. All submissions were peer-reviewed under the normal procedure of DCDS-B.
    Fred Wan has made influential and pioneering contributions in many areas of applied mathematics in his long career, such as shell and elastic theory, asymptotic analysis, and modeling and analysis of biological systems. The topics covered in this volume are mostly related to Fred's research on applications of mathematics and computations in mechanics and biology.

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Robustness of signaling gradient in drosophila wing imaginal disc
Jinzhi Lei Frederic Y. M. Wan Arthur D. Lander Qing Nie
Quasi-stable gradients of signaling protein molecules (known as morphogens or ligands) bound to cell receptors are known to be responsible for differential cell signaling and gene expressions. From these follow different stable cell fates and visually patterned tissues in biological development. Recent studies have shown that the relevant basic biological processes yield gradients that are sensitive to small changes in system characteristics (such as expression level of morphogens or receptors) or environmental conditions (such as temperature changes). Additional biological activities must play an important role in the high level of robustness observed in embryonic patterning for example. It is natural to attribute observed robustness to various type of feedback control mechanisms. However, our own simulation studies have shown that feedback control is neither necessary nor sufficient for robustness of the morphogen decapentaplegic (Dpp) gradient in wing imaginal disc of Drosophilas. Furthermore, robustness can be achieved by substantial binding of the signaling morphogen Dpp with nonsignaling cell surface bound molecules (such as heparan sulfate proteoglygans) and degrading the resulting complexes at a sufficiently rapid rate. The present work provides a theoretical basis for the results of our numerical simulation studies.
keywords: robustness nonlinear boundary value problem mathematical modeling. Morphogen gradient
Nonlinear three-dimensional simulation of solid tumor growth
Xiangrong Li Vittorio Cristini Qing Nie John S. Lowengrub
We present a new, adaptive boundary integral method to simulate solid tumor growth in 3-d. We use a reformulation of a classical model that accounts for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion. The 3-d method relies on accurate discretizations of singular surface integrals, a spatial rescaling and the use of an adaptive surface mesh. The discretized boundary integral equations are solved iteratively using GMRES and a discretized version of the Dirichlet-Neumann map, formulated in terms of a vector potential, is used to determine the normal velocity of the tumor surface. Explicit time stepping is used to update the tumor surface. We present simulations of the nonlinear evolution of growing tumors. At early times, good agreement is obtained between the results of a linear stability analysis and nonlinear simulations. At later times, linear theory is found to overpredict the growth of perturbations. Nonlinearity results in mode creation and interaction that leads to the formation of dimples and the tumor surface buckles inwards. The morphologic instability allows the tumor to increase its surface area, relative to its volume, thereby allowing the cells in the tumor bulk greater access to nutrient. This in turn allows the tumor to overcome the diffusional limitations on growth and to grow to larger sizes than would be possible if the tumor were spherical. Consequently, instability provides a means for avascular tumor invasion.
keywords: Solid tumor growth Nonlinear simulation 3D adaptive boundary integral method Morphological instability.
Robustness of Morphogen gradients with "bucket brigade" transport through membrane-associated non-receptors
Jinzhi Lei Dongyong Wang You Song Qing Nie Frederic Y. M. Wan
Robust multiple-fate morphogen gradients are essential for embryo development. Here, we analyze mathematically a model of morphogen gradient (such as Dpp in Drosophila wing imaginal disc) formation in the presence of non-receptors with both diffusion of free morphogens and the movement of morphogens bound to non-receptors. Under the assumption of rapid degradation of unbound morphogen, we introduce a method of functional boundary value problem and prove the existence, uniqueness and linear stability of a biologically acceptable steady-state solution. Next, we investigate the robustness of this steady-state solution with respect to significant changes in the morphogen synthesis rate. We prove that the model is able to produce robust biological morphogen gradients when production and degradation rates of morphogens are large enough and non-receptors are abundant. Our results provide mathematical and biological insight to a mechanism of achieving stable robust long distance morphogen gradients. Key elements of this mechanism are rapid turnover of morphogen to non-receptors of neighoring cells resulting in significant degradation and transport of non-receptor-morphogen complexes, the latter moving downstream through a "bucket brigade" process.
keywords: Morphogen gradient stability reaction diffusion equation functional boundary value problem. existence and uniqueness robustness
Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization
Zhenzhen Zheng Ching-Shan Chou Tau-Mu Yi Qing Nie
Cell polarization, in which substances previously uniformly distributed become asymmetric due to external or/and internal stimulation, is a fundamental process underlying cell mobility, cell division, and other polarized functions. The yeast cell S. cerevisiae has been a model system to study cell polarization. During mating, yeast cells sense shallow external spatial gradients and respond by creating steeper internal gradients of protein aligned with the external cue. The complex spatial dynamics during yeast mating polarization consists of positive feedback, degradation, global negative feedback control, and cooperative effects in protein synthesis. Understanding such complex regulations and interactions is critical to studying many important characteristics in cell polarization including signal amplification, tracking dynamic signals, and potential trade-off between achieving both objectives in a robust fashion. In this paper, we study some of these questions by analyzing several models with different spatial complexity: two compartments, three compartments, and continuum in space. The step-wise approach allows detailed characterization of properties of the steady state of the system, providing more insights for biological regulations during cell polarization. For cases without membrane diffusion, our study reveals that increasing the number of spatial compartments results in an increase in the number of steady-state solutions, in particular, the number of stable steady-state solutions, with the continuum models possessing infinitely many steady-state solutions. Through both analysis and simulations, we find that stronger positive feedback, reduced diffusion, and a shallower ligand gradient all result in more steady-state solutions, although most of these are not optimally aligned with the gradient. We explore in the different settings the relationship between the number of steady-state solutions and the extent and accuracy of the polarization. Taken together these results furnish a detailed description of the factors that influence the tradeoff between a single correctly aligned but poorly polarized stable steady-state solution versus multiple more highly polarized stable steady-state solutions that may be incorrectly aligned with the external gradient.
keywords: modeling. yeast pheromone gradient Cell polarization stability
Numerical methods for stiff reaction-diffusion systems
Ching-Shan Chou Yong-Tao Zhang Rui Zhao Qing Nie
In a previous study [21], a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear diffusion terms exactly and nonlinear reaction terms implicitly has excellent stability properties, and its second-order version, with a name IIF2, is linearly unconditionally stable. In this paper, we present another linearly unconditionally stable method that approximates both diffusions and reactions implicitly using a second order Crank-Nicholson scheme. The nonlinear system resulted from the implicit approximation at each time step is solved using a multi-grid method. We compare this method (CN-MG) with IIF2 for their accuracy and efficiency. Numerical simulations demonstrate that both methods are accurate and robust with convergence using even very large size of time steps. IIF2 is found to be more accurate for systems with large diffusion while CN-MG is more efficient when the number of spatial grid points is large.
keywords: morphogen gradients. Crank-Nicholson multi-grids reaction-diffusion equations Integration factor methods

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