Simulation of follicle formation by using the extended GM model.

<p>WNT activation pattern in wild type (a) and DKK+ transgenic mice (b) redrawn from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0171212#pone.0171212.ref030" target="_blank">30</a>]. Turing structures formed from homogenious noise in 5 minutes (internal time) with different levels of base DKK expression: 0.1, 0.15 and 0.2 <i>μ</i>M/s (c-d). Other parameters used are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0171212#pone.0171212.t001" target="_blank">Table 1</a>.</p

Morphogene adsorption as a Turing instability regulator: Theoretical analysis and possible applications in multicellular embryonic systems

<div><p>The Turing instability in the reaction-diffusion system is a widely recognized mechanism of the morphogen gradient self-organization during the embryonic development. One of the essential conditions for such self-organization is sharp difference in the diffusion rates of the reacting substances (morphogens). In classical models this condition is satisfied only for significantly different values of diffusion coefficients which cannot hold for morphogens of similar molecular size. One of the most realistic explanations of the difference in diffusion rate is the difference between adsorption of morphogens to the extracellular matrix (ECM). Basing on this assumption we develop a novel mathematical model and demonstrate its effectiveness in describing several well-known examples of biological patterning. Our model consisting of three reaction-diffusion equations has the Turing-type instability and includes two elements with equal diffusivity and immobile binding sites as the third reaction substance. The model is an extension of the classical Gierer-Meinhardt two-components model and can be reduced to it under certain conditions. Incorporation of ECM in the model system allows us to validate the model for available experimental parameters. According to our model introduction of binding sites gradient, which is frequently observed in embryonic tissues, allows one to generate more types of different spatial patterns than can be obtained with two-components models. Thus, besides providing an essential condition for the Turing instability for the system of morphogen with close values of the diffusion coefficients, the morphogen adsorption on ECM may be important as a factor that increases the variability of self-organizing structures.</p></div

Parameters of the extended GM model and their values fitted for reproducing Wnt/DKK hair follicle patterning.

<p>Parameters of the extended GM model and their values fitted for reproducing Wnt/DKK hair follicle patterning.</p

Turing instability for different values of <i>w</i>₀ and <i>μ</i>_<i>v</i>.

<p>Level lines of {max <b>Re</b> <i>λ</i>(<i>k</i>)} for different values of parameters and wave number (see supplementary equation (S1.3.9) for details). By definition, any vertical cross-section of the map gives a dispersion curve (similar to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0171212#pone.0171212.g002" target="_blank">Fig 2</a>). Dashed area indicates parameter values for which real parts of Lyapunov coefficients are positive and thus Turing instability can occur. The following model parameters were used for the development of the maps: <i>k</i>₁ = 1, <i>k</i>₋₁ = 0.1, <i>D</i> = 0.1, <i>ρ</i> = 0.6, <i>μ</i>_<i>u</i> = 0.05 and <i>w</i>₀ = 7 (a) or <i>μ</i>_<i>v</i> = 0.08 (b).</p

Reduction of the three-component model to the two-component one.

<p>A: Dispersion curves of the extended model (solid lines) and the conformable classical model (dashed line). <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0171212#pone.0171212.e024" target="_blank">Eq (7)</a> were used for the classical model. For each curve parameters of adsorption are presented under corresponding images <b>B</b>. Other parameters are fixed for both extended and classical models: <i>ρ</i> = 0.6, <i>μ</i>_<i>v</i> = 0.08, <i>μ</i>_<i>u</i> = 0.03. Reaction space of 20×20 space units and Neumann boundary conditions were used. Size of the reactor was set as 20 units. B: Concentration of activator as visualized after 4000 time units of the simulation with the above mentioned parameters.</p

Common shape of the dispersion curve.

<p>In the stationary state spatial structures (right pane) have a period which in many cases can be approximately predicted from the dispersion curve (left pane). Self-exited and damping spatial waves are shown with red and dashed lines, respectively.</p

The derivation of the extended Gierer-Meinhardt model.

<p>Kinetic scheme for the classical (a) and the extended (b) GM models are presented on the left side; the corresponding equations are on the right. Arrows indicating interactions between reactants have the same color as the reaction terms in corresponding equations: green for autocatalytic terms, red for inhibition of activator by inhibitor, gray for degradation terms and orange for the adsorption and the desorption terms.</p

Dispersion curves for various inhibitor degradation rates (a) and for various activator diffusion rates (b) in classical GM model.

<p>Dispersion curves are plotted using formula obtained in Eq (S1.2.6). Following parameters are used in both panels: <i>μ</i>_<i>u</i> = 1, <i>D</i>_<i>v</i> = 0.1. In panel (a): <i>D</i>_<i>u</i> = 0.001 and <i>μ</i>_<i>v</i> = 0.5, 5.4, 10.3, 15.3, 20.0 (colored by rainbow from red to cyan). In panel (b): <i>μ</i>_<i>v</i> = 5 and <i>D</i>_<i>u</i> = 0.001, 0.0018, 0.0032, 0.0056, 0.01 (colored in the same way).</p

Cellular uptake and intracellular distribution of the fluorescein-labeled mimic oligomers (TCACTCAACACTCAC-Flu)

<p><b>Copyright information:</b></p><p>Taken from "Hydroxyproline-based DNA mimics provide an efficient gene silencing and "</p><p>Nucleic Acids Research 2006;34(8):2247-2257.</p><p>Published online 2 May 2006</p><p>PMCID:PMC1456331.</p><p>© The Author 2006. Published by Oxford University Press. All rights reserved</p> () Fluorescent analysis of the delivery of oligomers (1 µM) into Phoenix Eco cells performed 20 h after the treatment under various delivery conditions: in the presence, or in the absence, of LFA, 6 mM CaCl, and 100 µM chloroquine (ClQ). () Confocal microscopy images of the pHypNA oligomer uptake after 20 h incubated with unfixed Phoenix Eco cells in the presence of: LFA (1); LFA/6mM Ca (2); CT-ODN/LFA (3) and LFA/100 µM ClQ (4)

Dose-dependent inhibition of firefly luciferase translation by the mimic antisense oligomers targeted against the translational start site of FLuc mRNA

<p><b>Copyright information:</b></p><p>Taken from "Hydroxyproline-based DNA mimics provide an efficient gene silencing and "</p><p>Nucleic Acids Research 2006;34(8):2247-2257.</p><p>Published online 2 May 2006</p><p>PMCID:PMC1456331.</p><p>© The Author 2006. Published by Oxford University Press. All rights reserved</p> () Sequences of HypNA-pPNA and pHypNA oligomers designed to target the FLuc mRNA; () Analysis of firefly luciferase translation inhibition by the oligomers. Firefly luciferase activity was calculated relative to the activity in the absence of any oligomer and normalized respect to luciferase production