Complex Dynamics of Osteoclast Formation and Death in Long-Term Cultures
Figure 7
Positive feedback on osteoclast formation is critical for obtaining oscillations with increasing amplitude.
A) Parametric portrait of the system in the space of parameters k9k10; k6–k8; k11. Bifurcation surface (A), described by equation (k6–k8–k11)2−4k9k10 = 0, separates the regions of non-oscillatory (below the surface), and oscillatory (above the surface) behavior. Bifurcation surface (B), described by equation k6–k8–k11 = 0, separates the regions of exponential growing and decaying behavior for osteoclasts and factor f. B) Increase in the value of parameter k6 results in development of oscillations of osteoclast numbers with increasing amplitude. C) Simultaneously increasing the value of parameter k6 and decreasing the value of parameter k5 allows simulation of experimental observation that the first peak is generally lower in experiments where osteoclasts oscillate with increasing amplitude compared to experiments without oscillations or with damped oscillations of osteoclasts. D) The 27 experiments where oscillations were observed were classified as damped oscillations, if the amplitude of the second peak was less then 50% of the first peak (n = 10), sustained oscillations, if the amplitude of the second peak was more than 50% but less then 150% of the first peak (n = 9), or unstable oscillations if the amplitude of the second peak was more then 150% of the first peak (n = 8), and the distance between 2 maximums was identified. Data are mean±SEM.