L 7: Linear Systems and Metabolic Networks

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L 7 Linear Systems and Metabolic Networks
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Linear Equations

• Form
• System

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Linear Systems
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Vocabulary

• If b0, then system is homogeneous
• If a solution (values of x that satisfy equation)
  exists, then system is consistent, else it is
  inconsistent.

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Solve system using Gaussian Elimination

• Form Augmented Matrix,
• Row equivalence, can scale rows and add and
  subtract multiples to transform matrix

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

• Vectors are linearly independent iff
• has the trivial solution that all the
  coefficients are equal to zero
• If mgtn, then vectors are dependent

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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 lt n

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Rank

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

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Inverse Matrix

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

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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 Ablb
• 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)

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Linear ODEs
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Steady-state Solution

• Under steady state conditions
• Need to find x

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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

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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

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Matrix Time Course

• Take a first order, linear, homogeneous ODE
• Solution is an exponential of the form
• General solution

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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

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Linearization

• Autonomous Systems does not explicitly depend on
  time (dx/dtf(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

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Linearization

• At steady state, look at deviation
• Use Taylors Expansion to approximate

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Linearization

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

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Stoichiometric Matrices

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

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Stoichiometric Network

• Matrix with m substrates and r reactions
• N nij is the matrix of size mxr

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External fluxes
Conventions are left to right and top down.
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Column/Row Operations

• System can be thought of as operating in row or
  column space

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Subspaces of Linear Systems