Figure 1.
Examples of bifurcation behavior.
(A) Bifurcation diagram of the system in Equation 1, an idealized model of a gene activated by signal as well as by its own protein product
, with parameters
,
,
. The three colored curves identify low, high, and unstable steady states for
(i.e., values for which
), as a function of the activating input
. Black arrows show the direction of change of
, assuming
constant. (B) With noise in the dynamics, individual cells would fluctuate in the vicinity of the steady states, leading to some overall distribution for
over time or across cells.
Figure 2.
Fluorescence data for the reporter protein indicating activity level of the galactose utilization network in S. cerevisiae.
Fluorescence is reported as a function of galactose level in culture (expressed as percent weight per volume; 1% = 10g/L), under the galactose pregrowth condition (A), and the raffinose pregrowth condition (B). All four biological replicates are shown stacked on each other. The blue area represents the number of cells counted in each fluorescence channel in replicate 1, the next lighter blue area is the sum of the counts in the first two replicates, and so on.
Figure 3.
Results of mixture modeling on replicate one.
(A) Means of subpopulations, as extracted by: mixture models estimated by the expectation-maximization algorithm (EM), mixture models estimated by a combination of mode estimation and expectation-maximization (ME+EM), and a conditional mixture model estimated by expectation-maximization (CEM). The x-axis represents the 17 levels of galactose tested, in order of increasing concentration. The y-axis represents fluorescence channels of the flow cytometer, which are proportional to the logarithm of fluorescent intensity. Darker background shading represents more cells counted in the channel at the given galactose level. (B) Estimated mixture coefficients (prior probabilities) of the low subpopulation as a function of galactose concentration. (C) Estimated standard deviations of the Gaussian distributions representing low (darker) and high (lighter) subpopulations as a function of galactose concentration.
Figure 4.
Comparison of subpopulation means and sizes across replicates.
(A) Subpopulation means as extracted by the three fitting methods, in all four replicates of the gal-pregrowth condition. (B) Subpopulation means in the four raf-pregrowth replicates. (C,D) Estimated sizes of the low subpopulations in the gal-pregrowth and raf-pregrowth conditions respectively.
Figure 5.
Emergence of the high subpopulation at increasing galactose concentrations, in the galactose pre-growth condition.
Empirical count distributions for the four replicates are shown, smoothed using a width-11 moving average to improve visibility. (A) At the third galactose concentration (0.0022%). (B) At the fourth galactose concentration (0.0033%). (C) At the fifth galactose concentration (0.0038%).
Figure 6.
Disappearance of the low subpopulation at higher galactose concentrations, in the galactose pre-growth condition.
Empirical count distributions for the four replicates are shown, smoothed using a width-11 moving average to improve visibility. (A) At the galactose concentration (0.0132%). (B) At the
galactose concentration (0.0152%). (C) At the
galactose concentration (0.0174%).
Figure 7.
Comparison of goodness-of-fit between methods and biological replicates.
(A) For each method, the mean negative log likelihood of the data. “Training” means each model is evaluated on the same data to which it is fit, whereas “testing” means each model is evaluated on the data from the other three replicates having the same pregrowth condition. Black bars indicate 95% confidence intervals. (B,C) Variability in the estimated locations of four subpopulations: the low (and only) subpopulation at the zero galactose concentration (P1), the low subpopulation at the galactose concentration (P2), the high subpopulation at the
galactose concentration (P3), the high (and only) subpopulation at the largest tested galactose concentration (P4). Cyan bars show the variability attributed to different estimation methods, whereas green bars show the variability attributed to different biological repliciates.