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.
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.
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 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.
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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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.
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.
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.
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.
In a previous study , 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.