L 7: Linear Systems and Metabolic Networks

Linear Equations Form System

Linear Systems

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.

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

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

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

Subspace of a vector space Defn: Subspace S of V n –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

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

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

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

Linear ODEs

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

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:

Effect of exponential power What happens for different values of a 11 ? Options: if system is perturbed –Stable- system goes to a steady-state –Unstable: system leaves steady-state –Metastable: system is indifferent

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

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

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

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

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

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

Stoichiometric Network Matrix with m substrates and r reactions N = {n ij } is the matrix of size mxr

External fluxes Conventions are left to right and top down.

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

Subspaces of Linear Systems