Biological cells continuously process stimuli that they receive from their inside and outside, using interconnected signaling pathways. Since strength, timing, and combination of inputs typically determine the output behavior in a non-linear manner, dynamic models of such signaling networks in the form of differential equations are becoming widely used to complement experimental studies 
. A major challenge to systems biology is therefore, to infer the rate constants, binding affinities and other parameters of such models from actual measurement data 
. Parameter estimation is a widely applied method for model calibration in mechanical and chemical engineering, and — if transferred to the field of cell signaling — may facilitate the development of predictive models of disease and therapy 
. However, because measurements are always afflicted with error, parameter estimates which minimize the discrepancy between model and data often come along with high uncertainty. Gutenkunst et al. 
studied parameter sensitivities in 17 published models in the field of systems biology, and concluded that parameter estimation in biology is difficult not only because of often large measurement error, but also because model behavior is often insensitive to combined changes of parameter values. The resulting working models thus remain limited in mechanistic insight and in their capability to predict system dynamics in unforeseen conditions.
For a given set of differential equations that describes a dynamic process, not every experimental protocol is equally suited to produce data from which parameters can be inferred. However, it is often possible to control a dynamic process in such a way that predicted measurements are very sensitive to the parameter values that underlie the dynamic system. In that case, the data would be better suited for parameter estimation because it restricts the range of parameter values that could explain the data. To efficiently determine experimental protocols that sensitize model predictions to parameter values, optimal control problems need to be solved. Körkel et al. 
developed and implemented an efficient scheme for solving such optimal control problems in chemical engineering, given deterministic models in the form of ordinary differential equations (ODE) or differential algebraic equations (DAE). These solutions propose how to best feed a reactor, control its temperature, and when to take measurements in order to estimate parameters with highest accuracy, while obeying constraints imposed by feasibility or cost. Although the computed experimental protocols and sampling schemes are unavoidably based only on approximate models and parameter values, experience shows that data evaluated from optimized experiments are typically more informative than data from intuitively planned experiments
. By designing experiments sequentially, the iterative process of performing experiments can be paralleled by an increasing predictive power to plan the next experiment.
Several computational studies on models of the MAP kinase cascade, apoptosis, STAT5 activation, and the
suggested that this strategy could also be valuable for the experiment-driven modeling of signal transduction networks. These studies showed that given a specific model of a biological process, several numerical methods can be used to identify stimulus and sampling patterns that would reveal parameter values with higher accuracy. The practical value of such an approach could be immense, because experiments are sensitized to reveal otherwise hidden biological parameters. However, despite a wealth of diverse numerical approaches 
(reviewed in 
), those methods have not been adopted by experimentalists. The theoretical nature of previous work left it unclear, whether optimized experiments would drive a true biological system within the predictive range of a model 
, how experimental uncertainty and imperfection affect the attainable accuracy of estimates, and how design performance is affected by the possibly large discrepancy between true biology and a differential equations model. Moreover, the increased complexity of proposed stimulus designs has raised the question whether numerically optimized experiments are even feasible enough to return net savings of experimental effort in a real-world cell biology lab.
Here, we addressed these questions explicitly by employing a numerical approach to enrich an existing single-cell assay for pharmacologically, biochemically, and clinically relevant parameters by an optimized protocol. Our study focused on the accumulation of the phosphatidylinositol 3,4,5-trisphosphate (
) second messenger lipid and the subsequent recruitment of downstream signaling elements through pleckstrin homology (PH) domains.
is produced by the catalytic subunit p110 of phosphoinositide 3-kinase (PI3K) in response to chemokine or other receptor stimuli in the plasma membrane of eukaryotic cells. Slow diffusion and rapid degradation of
result in gradients that are steep enough to mediate cell polarity as in migration and differentiation, but elevated synthesis of
is associated with cancer. Signaling elements downstream of PI3K, like the oncogenic Akt protein, engage by binding to
-enriched membranes through pleckstrin homology (PH) domains. Previously, Suh et al. 
described a chemical method for activating endogenous p110 at the plasma membrane. This technique makes use of the rapamycin-dependent heterodimerization of the FK506-binding protein (FKBP) with the mammalian target of rapamycin (mTOR). A genetic fusion of cyan fluorescent protein (CFP), FKBP12, and a peptide from the regulatory subunit p85 of PI3K was constructed (CF-p85) that resides in the cytosol and does not intrinsically stimulate p110 activity. However, upon addition of the rapamycin derivative iRap to cells co-transfected with a construct made from the N-terminal plasma membrane-targeting sequence of Lyn and the FKBP-rapamycin binding domain of mTOR (Lyn-FRB), CF-p85 translocates to the plasma membrane and induces the production of
(). Elevated concentrations of
can be monitored through translocation events of a construct in which yellow fluorescent protein is fused to the PH-domain of Akt (Y-PH).
Parameter estimation of PI3K signaling reveals identifiability problems for important kinetic properties.
Our experimental setup observed translocation dynamics of CF-p85 and Y-PH in NIH 3T3 fibroblasts by live-cell confocal fluorescence microscopy in order to infer the kinetic parameters of the underlying dynamic system. If such parameters were inferable with high statistical confidence, it would be possible, for example, to compare the inhibitory effects of different cancer drug candidates on PI3K directly and in situ
. To shed light on nonlinear signal transmission, it would be desirable also to reveal differential affinities of the more than 20 PH domains that were shown to be
. Moreover, the degradation of
is compromised in many tumors, whereas in others, sustained growth factor inputs could be responsible for elevated downstream signaling. We asked if the established single-cell assay would yield dynamic data from which such relevant properties could be inferred by fitting a differential equations model. However, we found that our intuitively planned and often used experimental protocol was not informative about these parameters. Even for a simplified model, the parameter estimates were largely undefined and covaried strongly. Since the traditional experimental protocol involved two drugs, iRap and the PI3K inhibitor LY294002 (LY29), we utilized a numerical optimization method 
to design better concentration-time profiles for these drugs in order to reveal model parameters with higher accuracy. We performed these numerically optimized experiments sequentially, and found that the uncertainty of parameter estimates could be reduced dramatically and covariance be largely eliminated. Given the potential implications on systems biology methodology, the paper derives meaningful constraints of the optimization problem as it pertains to live-cell experimentation, and investigates the implications of experimental uncertainty and imperfection.