Synthetic Biology and Stability Analysis in the Toggle Switch

TL;DR

Stability analysis of two-dimensional systems involves examining the eigenvalues of the system matrix and determining if their real parts are less than zero.

Transcript

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Key Insights

• 🥳 Stability analysis involves examining the eigenvalues of the system matrix and determining if their real parts are less than zero.
• 😥 The trace and determinant of the system matrix can be used to determine the stability of a fixed point in a two-dimensional system.

Questions & Answers

Q: How is stability determined in a one-dimensional system?

Stability in a one-dimensional system is determined by the sign of the derivative of the system. If it is negative, the fixed point is stable.

Q: How is stability determined in a two-dimensional system?

Stability in a two-dimensional system is determined by the trace and determinant of the system matrix. The trace must be negative, and the determinant must be positive for the fixed point to be stable.

Q: What are the possible behaviors of trajectories in a two-dimensional stable system?

Trajectories in a stable system can converge towards the fixed point along the eigenvectors.

Summary & Key Takeaways

• Stability in a one-dimensional system is determined by the sign of the derivative of the system. If it is negative, the fixed point is stable.

• In a two-dimensional system, stability is determined by the trace and determinant of the system matrix. The trace must be negative, and the determinant must be positive for the fixed point to be stable.

• Trajectories in a stable system converge towards the fixed point along the eigenvectors.