Supplementary to parameter estimation of dynamic biological network models using integrated fluxes

Transcription

1 Liu and Gunawan METHODOLOGY ARTICLE Supplementary to parameter estimation of dynamic biological network models using integrated fluxes Yang Liu and Rudiyanto Gunawan * * Correspondence: rudi.gunawan@chem.ethz.ch Institute for Chemical and Bioengineering, ETH Zurich, Vladimir-Perlog-Weg, 893 Zurich, Switzerland Full list of author information is available at the end of the article min X U,p I Φ X U, p I, X M s. t. X U p I L I, U I η t k. Simulate unmeasured X U X U = S U v X M, X U, p I ; X U = X U 5. Compute objective function Φ p I, X M. Calculate independent IFs t k η I,k X M, p I = v j X M, p I dt n m. Estimate the parameters associated with dependent IFs p D = argmin Φ in p D, η D, X M pd LD,UD 3. Solve for dependent IFs X M t k X M = S I S D η I,k X M, p I η D t k η D t k = S D X M t k X M S I η I,k X M, p I Figure S Flowchart of integrated flux parameter estimation (IFPE) with unmeasured concentrations. These unmeasured species are denoted by X U. All reactions and parameters appearing in ẊU are selected as independent fluxes and parameters. The first step in calculating Φ involves simulating unmeasured concentration X U by solving ODEs ẊU = S U v(x M, X U, p I ), where the measured concentrations X M are treated as external input variables. The subsequent steps are the same as in the IFPE methods without unmeasured concentrations. An example using the branched pathway model is shown in Figure S3. Modified Simpson s rule We performed the integration of flux function η(x, p) = t v(x, p)dt using the following modified Simpson s quadrature rule: t f(t)dt t t t 6 [ + 3β + β f(t ) + + 3β ] f(t ) β β( + β) f(t 3), () where β = (t 3 t )/(t t ). The quadrature function above was derived from the ordinary Simpson s rule, by calculating the shaded area of the quadratic polynomial determined by three points (t, f(t )), (t, f(t )), and (t 3, f(t 3 )) as illustrated

2 Liu and Gunawan Page of 5 in Figure S. In the special case that the time-series data are taken at equally distributed time points, the quadrature function above reduces to (by setting β = ): t t f(t)dt dt [5f(t ) + 8f(t ) f(t 3 )]. () To calculate the integral between last two time points, the area between t N and t N under the curve of the quadratic polynomial is tn f(t)dt t N t N t N 6 [ β ( + β ) f(t N ) + + 3β β f(t N ) + + ] 3β + β f(t N), (3) where β = (t N t N )/(t N t N ). Figure S Modified Simpson s rule Supplementary information for the branched pathway case study We generated the in silico data by simulating the ODE model in Eqs. () and () using the parameter values: a =, g 3 =.8, a = 8, g =.5, a 3 = 3, g 3 =.75, a = 5, g 3 =.5, g =., a 5 =, g 5 =.5, a 6 = 6, g 6 =.8, and the initial conditions: X (). X () X 3 () =.7.. X ().

3 Liu and Gunawan Page 3 of X 3 X.5.5 concentration X 3. X time Figure S3 Parameter estimation of the branched pathway model using noise-free data without the concentration measurements of X 3 using the IFPE without ODE integration. The solid lines and dots represent the model prediction and the true concentrations, respectively. (s, o) = (3, 3) (s, o) = (5, 3) (s, o) = (3, 5) concentration 3 5 (s, o) = (3, 3) 3 5 (s, o) = (5, 3) 3 5 (s, o) = (3, 5) time 3 5 Figure S Smoothened time-series concentration data using different piecewise spline-fitting parameters for the branched pathway case study. The parameters s and o indicate the number of pieces and the degree of polynomials, respectively. The plots in first row show noise-free data, while the plots in second row show one noisy dataset (out of five technical replicates) with % coefficient of variation of Gaussian noise.

4 Liu and Gunawan Page of 5 Table S Comparison of concentration errors Φ for the branched pathway case study Concentration error Noise-free data a Noisy data b (s, o) = (3, 3) (s, o) = (5, 3) (s, o) = (3, 5) (s, o) = (3, 3) (s, o) = (5, 3) (s, o) = (3, 5) SPE-slope ±.65.7 ±..7 ±. IPE-slope ±.3.79 ±..78 ±.8 IPE-ODE ±..6 ±..6 ±. SPE-ODE.8 3c.57 ±. IFPE.83.6 ±. IFPE-ODE ±. a. For noise-free data, five independent runs were carried out. The concentration error is reported for the run with the lowest objective function value. c. Only three out of five repeated runs finished within hours. The concentration error is reported for the run with the lowest objective function value among the three successful runs. Table S Comparison of median parameter errors for the branched pathway case study with different partitioning of independent and dependent flux set. Median parameter error a (%) Noise-free data b Noisy data c {V, V 6 } d ± ± 3.6 {V, V 5 } ± ± 6.9 {V, V 6 } ± ± 7.7 {V, V 3 } ± ± 5. {V, V 5 } ± ± 33.8 {V, V } ± ±.9 a. The median is taken over 3 parameters in the branched pathway model. b. For noise-free data, five independent runs were carried out. The median parameter error corresponds to the run with the lowest objective function value. c. For noisy data, the reported values are the mean ± standard deviation of five technical replicates d. Independent flux combination used in the main text. Table S3 Comparison of CPU times for the branched pathway case study with different partitioning of independent and dependent flux set. CPU time a (sec) Noise-free data b Noisy data c {V, V 6 } d ± ± 35 {V, V 5 } ± ± 3. {V, V 6 } ± ± 589. {V, V 3 } ± ± {V, V 5 } ±. 55. ± 39.8 {V, V } ± ± 99. a. The CPU times were recorded using a workstation with Intel Xeon processor 3.33GHz with 8GB RAM. b. For noise-free data, five independent runs were carried out. The CPU time is reported for the run with the lowest objective function value. c. For noisy data, the reported values are the mean ± standard deviation of five technical replicates d. Independent flux combination used in the main text. Table S Comparison of the number of ess iterations for the branched pathway case study with different partitioning of independent and dependent flux set. ess iterations Noise-free data a Noisy data b {V, V 6 } c ± ±.8 {V, V 5 } ± ±.9 {V, V 6 } ± ±. {V, V 3 } ± ± 69. {V, V 5 } ± ± 3.6 {V, V } ± ± 5. a. For noise-free data, five independent runs were carried out. The number of ess iterations corresponds to the run with the lowest objective function value. c. Independent flux combination used in the main text.

5 Liu and Gunawan Page 5 of 5 Table S5 Comparison of concentration error Φ for the branched pathway case study with different partitioning of independent and dependent flux set. concentration error Noise-free data a Noisy data b {V, V 6 } c ±..57 ±. {V, V 5 } ±.5.58 ±. {V, V 6 } ±..59 ±.3 {V, V 3 } ±.6.57 ±. {V, V 5 } ±..59 ±.3 {V, V } ±.5.57 ±. a. For noise-free data, five independent runs were carried out. The concentration error is reported for the run with the lowest objective function value. c. Independent flux combination used in the main text.