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Fig. 2 | BMC Evolutionary Biology

Fig. 2

From: Analysis of a mechanistic Markov model for gene duplicates evolving under subfunctionalization

Fig. 2

Pseudogenization rate \({h_{P}^{z}}(t)\) with for z=12 γ less than a, greater than b and equal to c γ crit. Panel d shows the overall shape of \({h_{P}^{z}}(t)\), with negative values of t included. a Pseudogenization rate \({h_{P}^{z}}(t)\) with \(\gamma < \gamma _{\text {crit}}^{z}\). The change in concavity occurs at t≈2.7. As such, the sigmoidal nature of the function is apparent - we see an initially slowly decreasing hazard rate which decreases more and more rapidly up to the change in concavity, after which the decline in the hazard rate slows, and approaches the asymptote at zero. b Pseudogenization rate \({h_{P}^{z}}(t)\) with \(\gamma > \gamma _{\text {crit}}^{z}\). Here the change in concavity occurs for some t<0, and hence cannot be seen in real, physical time (t>0). The shape is not obviously sigmoidal, and looks similar to that of an exponential decay. The rate of pseudogenization is initially declining rapidly, before approaching its asymptote at zero. c Pseudogenization rate \({h_{P}^{z}}(t)\) with \(\gamma = \gamma _{\text {crit}}^{z}\). Here the change in concavity occurs at exactly t=0. This is qualitatively similar to the case in panel b, with the pseudogenization rate rapidly declining, and the decline becoming slower as the rate approaches its asymptote at zero. d Pseudogenization rate \({h_{P}^{z}}(t)\) taken as a function over all \(\mathbb {R}\). This gives a complete picture of the shape of the pseudogenization rate function. Smaller values of γ translate the graph to the right, and result in a longer initial period of slowly declining pseudogenization rate. If γ>γ crit, the point of inflection occurs to the left of t=0, and we see behaviour similar to panel a

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