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ABSTRACT: In this paper, we consider a three-dimensional model of cell signal transduction. In this model, the deactivation of signalling proteins occur throughout the cytosol and activation is localized to specific sites in the cell. We use matched asymptotic expansions to construct the dynamic solutions of signalling protein concentrations. The result of the asymptotic analysis is a system of ordinary differential equations. This reduced system is compared to numerical simulations of the full three-dimensional system. As well, we consider the stability of equilibrium solutions. We find that the systems under consideration may undergo sustained oscillations, hysteresis and other complex behaviors. The simulations of the full three-dimensional system agree with simulations of the reduced ordinary differential equations.
Journal of Mathematical Biology 10/2012; · 2.96 Impact Factor
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ABSTRACT: When ceramics are heated inside a microwave cavity, a well-known phenomenon is the occurrence of hot spots – localised regions of high temperature. This phenomenon was modelled by Kriegsmann ((1997), IMA J. Appl. Math. 59(2), pp. 123–146; (2001), IMA J. Appl. Math. 66(1), pp. 1–32) using a non-local evolution PDE. We investigate profile and the stability of hot spots in one and two dimensions by using Kriegsmann's model with exponential non-linearity. The linearised problem associated with hot-spot-type solutions possesses two classes of eigenvalues. The first type is the large eigenvalues associated with the stability of the hot-spot profile and in this particular model there cannot be instability associated with these eigenvalues. The second type is the small eigenvalues associated with translation invariance. We show that the hot spots can become unstable due to the presence of small eigenvalues, and we characterise the instability thresholds. In particular, we show that for the material with low heat conductivity (such as ceramics), and in the presence of a variable electric field, the hot spots are typically stable inside a plate (in two dimensions) but can become unstable for a slab (in one dimension) provided that the microwave power is sufficiently large. On the other hand, for materials with high heat conductivity, the interior hot spots are unstable and move to the boundary of the domain in either one or two dimensions. For materials with moderate heat conductivity, the stability of hot spots is determined by both the geometry and the electric field inside the microwave cavity.
European Journal of Applied Mathematics 05/2011; 22(03):187 - 216. · 1.02 Impact Factor
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ABSTRACT: Intracellular signalling molecules form pathways inside the cell. These pathways carry a signal to target proteins which results in cellular responses. We consider a spherical cell with two internal compartments containing localized activating enzymes where as deactivating enzymes are spread uniformly through out the cytosol. Two diffusible signalling molecules are activated at the compartments and later deactivated in the cytosol due to deactivating enzymes. The two signalling molecules are a single link in a cascade reaction and form a self regulated dynamical system involving positive and negative feedback. Using matched asymptotic expansions we obtain approximate solutions of the steady state diffusion equation with a linear decay rate. We obtain three-dimensional concentration profiles for the signalling molecules. We also investigate an extension of the above system which has multiple cascade reactions occurring between multiple signalling molecules. Numerically, we show that the speed of the signal is an increasing function of the number of links in the cascade.
Journal of Mathematical Biology 01/2011; 63(5):831-54. · 2.96 Impact Factor
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SIAM Journal of Applied Mathematics. 01/2009; 69:1228-1243.
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ABSTRACT: In this paper, we consider a mathematical model for the formation 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 cros-sinhibition (modeled by Hill equations) to establish and maintain stable boundaries of gene expression. By computational analysis we find that in the presence of a general activator, neither autoactivation, nor cross-inhibition alone are sufficient 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.
Mathematical biosciences and engineering: MBE 05/2008; 5(2):277-98. · 1.13 Impact Factor
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Heidi Theisen,
Adeela Syed,
Baochi T Nguyen,
Tamas Lukacsovich,
Judith Purcell,
Gyan Prakash Srivastava, David Iron,
Karin Gaudenz,
Qing Nie,
Frederic Y M Wan,
Marian L Waterman,
J Lawrence Marsh
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ABSTRACT: Spatially restricted morphogen expression drives many patterning and regeneration processes, but how is the pattern of morphogen expression established and maintained? Patterning of Drosophila leg imaginal discs requires expression of the DPP morphogen dorsally and the wingless (WG) morphogen ventrally. We have shown that these mutually exclusive patterns of expression are controlled by a self-organizing system of feedback loops that involve WG and DPP, but whether the feedback is direct or indirect is not known.
By analyzing expression patterns of regulatory DNA driving reporter genes in different genetic backgrounds, we identify a key component of this system by showing that WG directly represses transcription of the dpp gene in the ventral leg disc. Repression of dpp requires a tri-partite complex of the WG mediators armadillo (ARM) and dTCF, and the co-repressor Brinker, (BRK), wherein ARM.dTCF and BRK bind to independent sites within the dpp locus.
Many examples of dTCF repression in the absence of WNT signaling have been described, but few examples of signal-driven repression requiring both ARM and dTCF binding have been reported. Thus, our findings represent a new mode of WG mediated repression and demonstrate that direct regulation between morphogen signaling pathways can contribute to a robust self-organizing system capable of dynamically maintaining territories of morphogen expression.
PLoS ONE 02/2007; 2(1):e142. · 4.09 Impact Factor
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ABSTRACT: We consider the following Schnakenberg model on the interval (-1,1): [formula see text] where D1 > 0, D2 > 0, B > 0. We rigorously show that the stability of symmetric N-peaked steady-states can be reduced to computing two matrices in terms of the diffusion coefficients D1, D2 and the number N of peaks. These matrices and their spectra are calculated explicitly and sharp conditions for linear stability are derived. The results are verified by some numerical simulations.
Journal of Mathematical Biology 11/2004; 49(4):358-90. · 2.96 Impact Factor
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ABSTRACT: We consider the following Schnakenberg model on the interval (Gamma1; 1): ! D 1 ? 0; D 2 ? 0; B ? 0: We rigorously show that the stability of symmetric N Gammapeaked steadystates can be reduced to computing two matrices in terms of the diffusion coefficients D 1 ; D 2 and the number N of peaks. These matrix and their spectrum are calculated explicitly and sharp conditions for linear stability are derived. 1.
06/2002;
