# Hybrid Dynamic Optimization Methods for Systems Biology with Efficient Sensitivities

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## Abstract

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## 1. Introduction

**Figure 2.**ErbB signaling pathway through a network of reactions with rates that are proportional to individual reactant concentrations [22]. This reaction graph is a simpler version of the ErbB signaling pathway than Chen et al. [20] (used in this study) due to the impracticality of displaying a larger model.

#### 1.1. Sequential Gradient-based Approaches

#### 1.2. Simultaneous Gradient-based Approaches

#### 1.3. Parameter Optimization

## 2. Simulation and Optimization of DAE Systems

#### 2.1. Simulation of DAE Systems

**Figure 4.**DAEs are discretized and solved over a time horizon with System States (x) determined from the equations and Inputs or Parameters (u) that may discontinuously adjust at points in the simulated time horizon.

#### 2.2. Optimization of DAE Systems

**Figure 5.**Overview of sequential (

**left**) and simultaneous (right) approaches to optimize DAE systems.

## 3. Hybrid Simultaneous and Sequential DAE Optimization

**Figure 6.**Overview of the hybrid approach to initialize and optimize DAE systems with initialization on the left followed by optimization on the right.

#### 3.1. Model Parameter Estimation with HIV Case Study

**Figure 7.**The parameters of the HIV model are adjusted to fit synthetic virus concentrations from simulated lab data.

**Figure 8.**Results of the HIV model parameter estimation, including predicted, actual, and measured virus (V), infected cells (I), and healthy cells (H).

**Figure 9.**Lower block triangular form with blocks of variables and equations that are solved successively and independently.

#### 3.2. Systems Biology: ErbB Signaling Pathway Model Parameter Estimation

#### 3.2.1. Improved Convergence through Model Transformation

#### 3.2.2. Solver Performance on 341 Benchmark Problems

**Figure 10.**Benchmark solver results for biological models comparing six solvers over 341 curated models from the biomodels database.

#### 3.2.3. Unobservable Parameters

#### 3.3. Results of ErbB Parameter Optimization

**Figure 11.**Lower block triangular form of ErbB model with original sparsity on the left and rearranged sparsity with successive blocks of variables and equations that are solved successively.

Parameters | Hybrid | Simultaneous |
---|---|---|

1 | 33.2 s | 30.6 s |

3 | 50.4 s | 45.4 s |

5 | 41.5 s | 67.8 s |

10 | 75.3 s | failure |

#### Post-Processing Analysis and Parameter Sensitivity

**Figure 13.**Parameter sensitivity over the time horizon (subplots

**a**and

**b**) and singular value magnitudes for principal parameter search directions (subplots

**c**and

**d**). The relative levels of the singular values show that some parameters have a strong effect on the predictions while others have a minimal effect or are co-linear and do not need to be estimated.

## 4. Discussion

## 5. Conclusions and Future Direction

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Lewis, N.R.; Hedengren, J.D.; Haseltine, E.L.
Hybrid Dynamic Optimization Methods for Systems Biology with Efficient Sensitivities. *Processes* **2015**, *3*, 701-729.
https://doi.org/10.3390/pr3030701

**AMA Style**

Lewis NR, Hedengren JD, Haseltine EL.
Hybrid Dynamic Optimization Methods for Systems Biology with Efficient Sensitivities. *Processes*. 2015; 3(3):701-729.
https://doi.org/10.3390/pr3030701

**Chicago/Turabian Style**

Lewis, Nicholas R., John D. Hedengren, and Eric L. Haseltine.
2015. "Hybrid Dynamic Optimization Methods for Systems Biology with Efficient Sensitivities" *Processes* 3, no. 3: 701-729.
https://doi.org/10.3390/pr3030701