Fig 1.
Flowchart for parameter analysis described in methods.
The polynomial fitting procedure is described in Algorithm 1. The parameter space samples are generated by model simulation. The sensitivity analysis and parameter estimation use the fitted surrogate polyomial.
Fig 2.
Error mean and standard deviation (measured using 100 random samples by cross-validation) for different polynomial fits (top), and the cost to compute the polynomials (bottom). (A) 5th order polynomials fit using different numbers of sample points. (B) Polynomials of varying degree using least squares fitting with 1000 points. Polynomial error is the average difference between the polynomial and the model output, and the error bars indicate the standard deviation of the error over the 100 sample points.
Table 1.
Sensitivities of output (Gbg/Gt) to kGa and kGd based on a 10th degree polynomial fit.
Sensitivity coefficients are given for different time points (T, secs) and α-factor concentrations (L, nM), as well as the overall mean.
Fig 3.
Parameter estimation of kGa and kGd in ODE model (1)–(4).
Probability distributions are obtained via Markov chain Monte Carlo and a 10th degree polynomial. (A) Distributions for individual parameters, normalized so that the total area is equal to 1. Red lines indicate the optimal (maximum likelihood) parameter values Popt. (B) Colormap of the two-dimensional joint probability distribution of kGa and kGd from the MCMC chain. Red indicates high probability along the diagonal; blue indicates low prsobability. (C, D) Model (blue) and polynomial (red) outputs corresponding to parameter sets P* and Popt, respectively, compared with the data (black) from [37] for the time-course (top) and dose-response (bottom) experiments.
Table 2.
Sensitivities of the ODE model output (Gbg/Gt) to all 8 kinetic parameters using a 5th degree surrogate polynomial.
Both mean sensitivities and the mean of the absolute value of the sensitivities are shown.
Fig 4.
MCMC results for ODE model (1)–(4).
(A) Parameter distributions from ODE model (1)–(4) for all 8 kinetic parameters obtained via MCMC. Red vertical lines indicate the parameter values from experiments or maximum likelihood estimates [37]. Markov chain length was 106 steps. (B) Model simulation and polynomial outputs using the mean parameter set from the 8-parameter MCMC compared with the time-course (top) and dose-response (bottom) data.
Table 3.
Parameter estimates and ranges from previous work.
SA denotes cell surface area and V denotes cell volume. The distance unit is μm, the time unit is seconds, and concentration is measured as the number of molecules per unit surface area or volume (unless otherwise specified).
Table 4.
Sensitivity coefficients, in order of ascending magnitude, for the reduced 15-parameter PDE model based on a 5th order polynomial fit using 6,000 sample points.
Fig 5.
Parameter distributions based on MCMC with chain length 2 × 106 for the reduced 15-parameter PDE model. The parameter range is a log-scale except for the parameters q and h which span a linear scale.
Fig 6.
MCMC vs. optimization results.
Steady state solutions for the mean parameter set from the MCMC (solid black) and a parameter set identified via simulated annealing (dashed black). Polarization is depicted by the concentration of active Cdc42 (C42a, number of molecules/μm2) over the angular range [−π, π]. The mean polarization of the experimental data (n = 20 cells) is shown in blue in arbitrary units. A sample cell, treated with 10 nM α-factor for 60 min, shows the membrane polarization profile of Ste20-GFP, a reporter for active Cdc42 (upper left). Scale bar = 2 μm.