Optimization of an amplification protocol for misfolded proteins by using relaxed control.

Area of Science:

• Biophysics
• Mathematical Biology
• Control Theory

Background:

• Misfolded proteins are implicated in prion diseases.
• Current amplification techniques require optimization for disease detection.
• Optimal control problems are crucial for refining biological processes.

Purpose of the Study:

• To investigate an optimal control problem for amplifying misfolded proteins.
• To enhance techniques for prion disease detection.
• To understand and overcome challenges in optimal control for biological systems.

Main Methods:

• Modeling the system using linear differential equations with nonlinear control.
• Applying Perron and Floquet eigenvalue theory to analyze oscillations in numerical simulations.
• Relaxing the optimal control problem to address unsolvability.
• Explicitly solving the two-dimensional optimal relaxed control problem for large final times.

Main Results:

• Oscillations in simulations were explained using Perron and Floquet eigenvalue theory.
• A method to overcome the unsolvability of the optimal control was developed.
• The optimal relaxed control problem was explicitly solved in a specific two-dimensional case.

Conclusions:

• The study provides a mathematical framework for optimizing protein amplification techniques.
• The findings contribute to improved methods for detecting prion diseases.
• Relaxation of optimal control problems offers a viable approach for complex biological systems.