Optimal experimental design for parameter estimation of a cell signaling model

Abstract

Differential equation models that describe the dynamic changes of biochemical signaling states are important tools to understand cellular behavior. An essential task in building such representations is to infer the affinities, rate constants, and other parameters of a model from actual measurement data. However, intuitive measurement protocols often fail to generate data that restrict the range of possible parameter values. Here we utilized a numerical method to iteratively design optimal live-cell fluorescence microscopy experiments in order to reveal pharmacological and kinetic parameters of a phosphatidylinositol 3,4,5-trisphosphate (PIP(3)) second messenger signaling process that is deregulated in many tumors. The experimental approach included the activation of endogenous phosphoinositide 3-kinase (PI3K) by chemically induced recruitment of a regulatory peptide, reversible inhibition of PI3K using a kinase inhibitor, and monitoring of the PI3K-mediated production of PIP(3) lipids using the pleckstrin homology (PH) domain of Akt. We found that an intuitively planned and established experimental protocol did not yield data from which relevant parameters could be inferred. Starting from a set of poorly defined model parameters derived from the intuitively planned experiment, we calculated concentration-time profiles for both the inducing and the inhibitory compound that would minimize the predicted uncertainty of parameter estimates. Two cycles of optimization and experimentation were sufficient to narrowly confine the model parameters, with the mean variance of estimates dropping more than sixty-fold. Thus, optimal experimental design proved to be a powerful strategy to minimize the number of experiments needed to infer biological parameters from a cell signaling assay.

Figure 1. Parameter estimation of PI3K signaling reveals identifiability problems for important kinetic properties.

(A) iRap-induced dimerization of FKBP with FRB recruits CF-p85 to the plasma membrane and activates the catalytic subunit p110 of PI3K to produce from . PH domains of many downstream signaling proteins bind to enriched membranes with differential affinities. LY29 is a widely used inhibitor of PI3K. (B) Confocal fluorescence images of an NIH 3T3 cell (CF-p85 shown in cyan, Y-PH in yellow). Initially, CF-p85 and Y-PH reside in the cytosol (left). Addition of iRap recruits both constructs to the plasma membrane as can be judged from cytosolic depletion (middle). Y-PH but not CF-p85 dissociates from the plasma membrane upon addition of LY29. Experiment time is shown in minutes. (C) Differential equations model of the dynamic system illustrated in (A). System parameters are shown in blue, variables that allow control of the system are shown in green, and calibration parameters which were different from cell to cell are shown in gold. (D) Parameter values can be inferred by minimizing the discrepancy between model and experimental data. The schematic illustrates this concept for two parameters, and . The range of possible parameter values is where this discrepancy falls below the confidence level (blue shaded ellipsoid). If this range is large, parameters can be inferred only with high uncertainty. (E) Fluorescence was quantified in the green square depicted in (B), and the model in (C) was fit to that data. Error bars show data, continuous lines show model output. Blue error bars and lines show cytosolic fluorescence of CF-p85, orange error bars and lines cytosolic fluorescence of Y-PH. (F) The relative uncertainty of parameter estimates was very high. This is a common problem in systems biology models. (G) Schematic projection of the confidence ellipsoid from (D). Good experiments minimize the size of this uncertainty region (shaded blue). Criteria that have been proposed for describing this size are the longest half-axis, the average extent, or the volume of the confidence ellipsoid, among others. These criteria correspond to properties of the parameter covariance matrix .

Figure 2. Numerical optimization of experimental design and practical experimentation to infer model parameters from live-cell microscopy data.

(A) Schematic of sequential experimentation. The traditional design (left) resulted in a model that was used for optimizing the concentration-time profile of iRap (green) and LY29 (pink) for the next experiment. (B) The washout of LY29 at minute 15 was realized by inserting a needle connected to a vacuum trap, and rinsing with 5 ml of iRap in extracellular buffer. (C) Mean of parameter variances predicted for or achieved with the optimized experiment. Arrows indicate the parameter variance at different stages of model development, and under consideration of experimental uncertainty introduced by the wash. (D) Confocal fluorescence images of an NIH 3T3 cell (CF-p85 shown in cyan, Y-PH in yellow). 30 nM iRap was added immediately before triggering time-lapse acquisition. After 5 min, iRap and LY29 were added. 10 min later, LY29 was washed out, using iRap in extracellular buffer. Experiment time is shown in minutes. (E) Combined parameter estimation from both the traditional (left) and optimized (right) experiment. Error bars show data, continuous lines show model output. Blue error bars and lines show cytosolic fluorescence of CF-p85, orange error bars and lines cytosolic fluorescence of Y-PH. The frame captured at minute 15 of the optimized experiment was lost to brightfield illumination during the washout procedure.

Figure 3. Minimization of different uncertainty criteria under experimentally meaningful design constraints.

Different uncertainty criteria were minimized by finding optimal time points for adding drugs and optimal concentrations. Washouts were excluded from the design space by a dynamic constraint formulation. (A) Performance comparison of different criteria with respect to parameter uncertainty or (B) the eigenvalue spectrum of the optimized experimental design. Arrows indicate the eigenvalue that is aligned with the uncertainty of . (C) Schematic of sequential experimentation. The data from the traditional and the first optimized experiment were used for optimizing the concentration-time profile of iRap (green) and LY29 (pink) for the next experiment. (D) NIH 3T3 cell (CF-p85 shown in cyan, Y-PH in yellow). iRap was added at the beginning of the experiment. 10 min later, another 10 nM LY29 and another iRap were added. PI3K remained active as judged from low cytosolic fluorescence of Y-PH. LY29 was added at minute 20, resulting in dissociation of Y-PH from the plasma membrane. Time is shown in minutes. (E) Combined parameter estimation on all three experiments. Error bars show data, continuous lines show model output. Blue error bars and lines show cytosolic fluorescence of CF-p85, orange ones show cytosolic fluorescence of Y-PH.

Figure 4. Decline of uncertainty.

(A) The mean of parameter variances dropped from 193.5 after the traditional experiment to 2.9 after the second optimized one. For comparison, parameter uncertainties were calculated for repetitions of the intuitive traditional experiment. (B) Parameter uncertainties after individual experiments were performed. (C) Normalized eigenvectors of the corresponding covariance matrices ordered in ascending length suggest that correlation between biological parameters was largely eliminated. (D) The ranges of the eigenvalue spectra of were consistently reduced.