Supplementary Figure 9: A kinetic model captures transition rates among the three topological states.

a, A series of first-order reactions are used to model the transition kinetics between the O_off, P_off and P_on states. Based on the finding that physical proximity is required for transcriptional activation, we assume in this model that P_on occurs only after P_off is established. The coupled ODEs describe evolution of the system given the initial conditions. For parS-homie-noPr-PP7, only the O_off and P_off states are present and we assume the same f₁ and b₁ values as for parS-homie-evePr-PP7. b, Fraction of the P_off state for homie-noPromoter-PP7 as a function of developmental time. 0 on the x axis corresponds to 25 min in nc14. The mean ± SE is shown (n = 29 embryos). This curve, together with time series curves obtained from the parS-homie-evePr-PP7 construct (dashed lines; same as in Fig. 3e), is used to infer the kinetic parameters with Markov chain Monte Carlo (MCMC) simulations (Methods). c–g, Marginal posterior distributions of the five kinetic parameters in a constructed from 90,000 stationary MCMC samples. Medians are labeled. Error bars span from the 25th-percentile to the 75th-percentile quantile (also shown in square brackets). Insets in f and g show the joint distribution of (b₁, b₂) and (b₁, b₃), respectively. Darker color represents higher density. h, The inferred parameters for the disappearance of P_on recapitulate the distribution of lifespans of PP7 activity. To calculate PP7 lifespan distribution, PP7 traces are grouped into cohorts according to the maximum measurable lifespan for each trace (Methods). For each PP7 cohort, a cumulative distribution function (CDF) for the PP7 lifespan is calculated (gray curves). Because the lifespan distribution is truncated at the maximum measurable time, the tails of the CDFs (corresponding to CDF = 1) are removed. The solid red line shows the median of these truncated CDFs, which is the CDF of the lifespans of PP7 activity. The dashed red curve comes from the CDF of an exponential distribution with mean = (b₂ + b₃)^–1 = (0.014 + 0.011)^–1 min. This exponential CDF is shifted horizontally to account for a deterministic elongation time of 4 min, which coincides with the lifespan of the shortest PP7 trace.