Figure 1.
Generalized illustration of the TCR-mediated signaling pathway through the Erk-MAPK cascade.
Arrow- and diamond-heads denote activation and inhibition of substrate molecules, respectively. TCR stimulation is achieved through αCD3 (green) binding. Subsequent Erk activation (black) controlled using small molecule inhibitors sanguinarine (blue) and U0126 (red).
Figure 2.
A framework for multiple-model open-loop control of uncertain intracellular signaling in the laboratory.
First, the model bank is populated with a set of relevant models that predict the system response to possible control inputs. During the initial stage, training data are used to generate weights maps. These maps inform the controller of the tendency for models to differ in their ability to accurately reproduce the system dynamics under different control inputs. During each time interval of the controller stage, the performance metrics for the models are optimized simultaneously using a multiobjective technique within a MPC framework to generate a candidate solution set. The tasks involved in the adaptive model weighting strategy are contained within the gray box: control inputs are selected from the solution set by prioritizing them according to the weight maps, then model weights are automatically recalibrated using the portion of training data that most closely corresponds to the proposed control input. Optimization and input selection cycles repeat for subsequent time intervals as the prediction horizon slides along until the entire open-loop control sequence is specified and ready to be applied to the in vitro system.
Figure 3.
Simulations indicate adaptive weighting strategy significantly improves overall target tracking performance.
(A–E) Control input dosing regimens for the matched (SK) and mismatched (SZ and SL) single-model controllers and the multiple-model controllers with fixed equal weights (Meq) and with adaptive Akaike weights (Maw). (F) Akaike weights for Maw. (G) Target trajectory (solid black) and simulated system (Model K) responses controlled by SK (dashed red), SZ (dotted blue), SL (dotted green), Meq (dashed cyan) and Maw (solid magenta). (H) The squared error values for all five controllers.
Figure 4.
Summary of target tracking performances for the simulated case studies.
(A) Summary of experiments involving realistic control reagents sanguinarine and U0126. (B) Summary of experiments involving hypothetical reagents aZAP and iZAP. Target tracking performance is measured by the squared error between target profiles and controlled plants. Data shown are mean ± standard error between matched (Sm, n = 30) and mismatched (Smis, n = 60) single-model controllers, Meq (n = 30) and Maw (n = 30). Group letters denote statistically significant differences between groups (p<0.05) as calculated by one-way ANOVA with Tukey multiple comparisons test (SigmaStat v3.5, Systat Software, Inc).
Figure 5.
In vitro experiments demonstrate superior target tracking performance by Maw; corroborates observed in silico trends.
(A–D) Control input dosing regimens for single-model controllers (SZ, SL and SK) and the multiple-model controller with adaptive Akaike weights (Maw). (E) Akaike weights for Maw.
Figure 6.
Overall in vitro target tracking performances between target profiles and measured Erk phosphorylation.
Experiments with target trajectories (solid black) defined by the (toff, pss) pairs of (A) (8,0), (B) (15,0) and (C) (22,0). Data are measurements of plant dynamics that were uncontrolled (UC, cyan triangle, n = 12) and controlled by SZ (blue square, n = 9), SL (green x, n = 9), SK (red circle, n = 9) and Maw (magenta dot, n = 9). Data shown are mean ± standard error. (D) Controller performances as measured by squared error between target trajectories and controlled plant dynamics. Group letters denote statistically significant differences between groups (p<0.05) as calculated by one-way ANOVA with Tukey multiple comparisons test (SigmaStat v3.5, Systat Software, Inc).
Figure 7.
Analysis of uncertainty in the model weight maps and resulting predicted control strategies and performance.
Model weights over the input space for (A–C) the original case study (full dataset) and (D–F) the case study with the limited dataset. Model rankings over the input space for (G–I) the original case study (full dataset) and (J–L) the case study with the limited dataset. (M) Adaptive weights and (N) corresponding control input regimen for the exemplar experiment characterized by the target trajectory defined by the pair (toff = 8, pss = 0). (O) Controller performances as measured by squared error between target trajectories and plant dynamics controlled by single-model controllers that are matched (Sm, n = 30) and mismatched (Smis, n = 60), and adaptively-weighted multiple-model controllers with the original “digital” weight maps (Maw(o), n = 30) and smoother weight maps from the limited dataset (Maw(ld), n = 30). Data shown are mean ± standard error. Group letters denote statistically significant differences between groups (p<0.05) as calculated by one-way ANOVA with Tukey multiple comparisons test (SigmaStat v3.5, Systat Software, Inc).
Figure 8.
Training data used to generate the model weight maps.
Rows correspond to doses of sanguinarine only, U0126 only, and combinations of the two reagents, respectively. Symbols and error bars denote means ± standard errors of the raw normalized data. Lines represent smoothed data. Arrows denote the time at which the reagents were administered.
Figure 9.
Illustration of control input selection with model weight adaptation.
For a given prediction time interval in the control process, Pareto-optimal control inputs are computed by the multi-model MPC strategy by solving (7), which then enter an iterative process of control input selection and weight adaptation. (A) First, Pareto points are ranked with an initial weight vector ω0 and the optimal point (example: black square) is selected using (8). (B) Next, the input vector corresponding to the selected optimal point (u1, black square) is identified. If the input vector continues to change above a pre-defined threshold, the process continues to the next iteration. (C) Given the current input vector (u1), a new weight vector (ω(u1), black square) is computed. The process continues and repeats (example: black circle, then magenta star) until the aforementioned stopping criterion is met. The final input vector (un, magenta star) is returned to the main control loop as the best compromise control strategy and used to update the prediction models in preparation for the next prediction time interval.