L 7: Linear Systems and Metabolic Networks

1. L 7: Linear Systems and Metabolic Networks

2. Linear Equations • Form • System

3. Linear Systems

4. Vocabulary • If b=0, then system is homogeneous • If a solution (values of x that satisfy equation) exists, then system is consistent, else it is inconsistent.

5. Solve system using Gaussian Elimination • Form Augmented Matrix, • Row equivalence, can scale rows and add and subtract multiples to transform matrix

6. Overdetermined: fewer unknowns, than equations, if rows all independent, then no solution Underdetermined: more unknowns, than equations, multiple solutions

7. Linear Dependency • Vectors are linearly independent iff • has the trivial solution that all the coefficients are equal to zero • If m>n, then vectors are dependent

8. Subspace of a vector space • Defn: Subspace S of Vn • Zero vector belongs to S • If two vectors belong to S, then their sum belongs to S • If one vector belongs to S then its scale multiple belongs to S • Defn: Basis of S: if a set of vectors are linearly independent and they can represent every vector in the subspace, then they form a basis of S • The number of vectors making up a basis is called the dimension of S, dim S < n

9. Rank • Rank of a matrix is the number of linearly independent columns or rows in A of size mxn. Rank A < min(m,n).

10. Inverse Matrix • Cannot divide by a matrix • For square matrices, can find inverse • If no inverse exists, A is called singular. • Other useful facts:

11. Eigenvectors and Eigenvalues • Definition, let A be an nxn square matrix. If l is a complex number and b is a non-zero complex vector (nx1) satisfying: Ab=lb • Then b is called an eigenvector of A and l is called an eigenvalue. • Can solve by finding roots of the characteristic equation (3.2.1.5)

12. Linear ODEs

13. Steady-state Solution • Under steady state conditions • Need to find x:

14. Time Course • Take a first order, linear, homogeneous ODE: • Solution is an exponential of the form • Put into equation, solve for constant using ICs gives:

15. Effect of exponential power • What happens for different values of a11? • Options: if system is perturbed • Stable- system goes to a steady-state • Unstable: system leaves steady-state • Metastable: system is indifferent

16. Matrix Time Course • Take a first order, linear, homogeneous ODE: • Solution is an exponential of the form • General solution:

17. Why are linear systems so important? • Can solve it, analytically and via computer • Gaussian Elimination at steady state • Properties are well-known • BUT: world is nonlinear, see systems of equations from simple systems that we have already looked at

18. Linearization • Autonomous Systems: does not explicitly depend on time (dx/dt=f(x,p)) • Approximate the change in system close to a set point or steady state with a linear equation. • It is good in a range around that point, not everywhere

19. Linearization • At steady state, look at deviation: • Use Taylor’s Expansion to approximate:

20. Linearization • First term is zero by SS assumption, assume H.O.T.s are small, so left with first order terms

21. Stoichiometric Matrices • Look at substances that are conserved in system, mass and flow • Coefficients are proportion of substrate and product

22. Stoichiometric Network • Matrix with m substrates and r reactions • N ={nij} is the matrix of size mxr

23. External fluxes Conventions are left to right and top down.

25. Column/Row Operations • System can be thought of as operating in row or column space

26. Subspaces of Linear Systems