70 research outputs found

    Two-dimensional diagrams showing the dependence of amplitude of oscillations of the models with monotonic synthesis with and , along with bifurcation curves in blue.

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    <p><b>A:</b> the model with saturable degradation. <b>B:</b> the model with non-saturable degradation. Notice that the amplitude of the model with saturable degradation increases with , whereas the amplitude of the model with non-saturable degradation saturates. for both models, for the model with saturable degradation, and for the model with non-saturable degradation. and are chosen to keep the equilibrium state at .</p

    Various time series of the four models.

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    <p><b>A:</b> Time series for the models with monotonic synthesis at . The model with saturable degradation is in red, and the model with non-saturable degradation is in green. <b>B:</b> Time series for the model with non-monotonic synthesis and saturable degradation at in red and at in green. Our simulations indicate that the models with monotonic synthesis stay robust at high and , whereas the models with non-monotonic synthesis become chaotic at high and . for all models, for the models with saturable degradation, and for the model with non-saturable degradation. and chosen to keep the equilibrium state at .</p

    Phase portraits of the two models with monotonic synthesis, along with their second-order reductions to systems of ODEs, at .

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    <p><b>A:</b> the model with saturable degradation. <b>B:</b> the model with non-saturable degradation. In both figures, the red curve is the original model, the green curve is the second-order reduction, and the blue dotted and black curves are switching curves. The closeness with which the second-order reductions approximate the originals shows that the second-order reduction technique is valid. Note that the -axes for the two graphs are different for better resolution. for all models, for the models with saturable degradation, and for the model with non-saturable degradation. and chosen to keep the equilibrium state at .</p

    Time series of the two models with monotonic synthesis, along with their first- and second- order reductions, at and two different values of ( for A, for B).

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    <p><b>A:</b> the model with saturable degradation. <b>B:</b> the model with non-saturable degradation. In both figures, the red curve is the original model, the green curve is the first-order reduction, and the blue curve is the second-order reduction. For both models, both reductions approximate the originals well. However, the periods of the first-order reductions are slightly off from the originals, whereas the periods for the second-order reductions are much closer. for all models, for the models with saturable degradation, and for the model with non-saturable degradation. and chosen to keep the equilibrium state at .</p

    Two-dimensional diagrams showing the dependence of the correlation dimension of the time series obtained from the models with non-monotonic synthesis on and , along with bifurcation curves in blue.

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    <p><b>A:</b> the model with saturable degradation. <b>B:</b> the model with non-saturable degradation. The diagrams indicate that for high and , the models with non-monotonic synthesis exhibit high-dimensional, chaotic behavior. Note that the color axes vary between the two diagrams. for all models, for the models with saturable degradation, and for the model with non-saturable degradation. and chosen to keep the equilibrium state at .</p

    Diagrams showing the dependence of the numerically-calculated dimension of the trajectory of the models with non-monotonic synthesis on , along with the analytical dimension lines, for different .

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    <p>The set of diagrams on the top corresponds to the model with saturable degradation, and the set of diagrams on the bottom corresponds to the model with non-saturable degradation. The diagrams indicate that the slope of the analytical dimension lines match the slope of the numerically-calculated dimension points. It is important to note that the numerical estimates fail for high dimension, as evidenced by the trailing points in the bottom set of diagrams. The analytical dimension lines have no such limitation. for all models, for the models with saturable degradation, and for the model with non-saturable degradation. and chosen to keep the equilibrium state at .</p

    A graph of the synthesis terms near for varying .

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    <p><b>A:</b> The monotonic synthesis term . Because the term is monotonically decreasing, it represents universal negative feedback. Furthermore, as increases, becomes increasingly step-like. <b>B:</b> The non-monotonic synthesis term . Because the term is not monotonically decreasing, it represents feedback that switches from positive to negative near . We have chosen to scale both graphs to by setting to .</p

    PORCN regulates a subset of genes in a Wnt-independent manner.

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    <p>A. Gene expression analysis of MDA-MB-231 cells 48 hours after mock transfected (NT) or transfected with 100 nM of indicated siRNA for 48 hrs. PORCN and WLS differentially regulate a distinct subset of genes. B. To illustrate genes differentially regulated by PORCN, the abundance of the indicated transcripts were assessed by qRT-PCR 48 hours after transfection of MDA-MB-231 cells with the indicated siRNAs C. Two independent PORCN siRNAs cause decrease in APEH protein. Immunoblot analysis of APEH and Actin from MDA-MB-231 cells treated with 100 nM Control (C), PORCN7 (P7) or PORCN8 (P8) siRNA for 72 hrs. D. PORCN knockdown triggers loss of APEH in multiple cell lines. Immunoblot analysis of APEH from MCF7, T47D, and DLD-1 cells transfected with 100 nM Control (C), or PORCN7 (P7) siRNAs for 72 hrs. Top panel is 10 s exposure, middle panel is 30 s exposure. Actin (bottom panel) serves as a load control. E. Loss of APEH is blocked by RNAi-immune wildtype and mutant PORCN. Immunoblot analysis of APEH and HA-tagged PORCN in MDA-MB-231 cells co-transfected with P7-resistant WT or H341A PORCN and 100 nM Control (C) or PORCN7 (P7) siRNA for 48 hrs. * = non-specific band recognized by anti-HA antibody, used as load control. Experiment done in duplicate with same results. F. Model of PORCN's function in transformed epithelial cells. PORCN is involved in the acylation and secretion of Wnt proteins. Independent of its role in Wnt secretion, PORCN regulates proliferation and gene expression. The role in gene expression may lead to changes in proliferation.</p
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