Linear Algebraic Systems II

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Linear Algebraic Systems II

• Vector and matrix norms
• Generalized matrix problems
• Least-squares minimization
• Metabolic flux balance analysis

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Vector Norms

• A scalar measure of vector magnitude
• Notation x
• Common norms
• Example xT 2 -3 0 1 -4

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Vector Norms cont.

• Properties
• x is a non-negative real number
• x 0 if and only if x 0
• kx k x
• x y lt x y (triangular
  inequality)
• Example Euclidean norm
• First three properties hold trivially
• Triangular inequality can be proved (see text)

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Matrix Norms

• A scalar measure of square matrix magnitude
• Notation A
• Equivalent definitions
• Common norms

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Matrix Norms cont.

• Properties
• Satisfy same four properties as vector norm
• Ax lt A x, AB lt A B
• Example

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Condition Number

• Definition k(A) A A-1
• A large condition number indicates an
  ill-conditioned matrix
• Well conditioned matrix
• Ill-conditioned matrix

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Condition Number cont.

• Effect of data errors
• Small data errors can lead to large solution
  errors
• Bound can be very conservative
• Example

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Generalized Linear System Solutions

• Linear algebraic system
• Completely defined system m n
• Equal number of equations unknowns
• Unique solution exists if det(A) is non-zero
• Overdetermined system m gt n
• More equations than unknowns
• No general solution exists
• Underdetermined system m lt n
• More unknowns than equations
• Infinite number of solutions exist

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Overdetermined Systems

• Problem Ax b
• Not enough unknowns to satisfy all equations
• Find Ax closest to b
• Common problem in parameter estimation
• Minimize least-squares metric
• Minimization
• Matrix differentiation formula

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Overdetermined Systems cont.

• Normal equations (ATA)x ATb
• Solution x (ATA)-1ATb
• ATA must be nonsingular
• (ATA)-1AT is called the left inverse matrix
• Example

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Underdetermined Systems

• Problem Ax b
• Too many unknowns
• Find smallest x that satisfies equations
• Minimize least-squares metric
• Minimization problem
• Convert constrained problem into unconstrained
  problem
• Utilize method of Lagrange multipliers (see text)
• Solution x AT(AAT)-1b
• AAT must be nonsingular
• AT(AAT)-1 is called the right inverse

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Underdetermined Systems cont.

• Example

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Other Matrix Differentiation Formulas

• Linear functions
• Quadratic functions

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Intracellular Mass Balances

• Stoichiometric coefficients (a1,a2)
• Determines number of product molecules produced
  per molecule of substrate consumed
• Often known from biochemistry literature
• Reaction rates (v1,v2)
• Also called reaction velocity or flux
• Can be determined experimentally or
  computationally
• Metabolic flux balance analysis
• Calculation of unknown fluxes from available
  measurements

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Yeast Glycolysis
Wolf Heinrich, Biochemical Journal, 345,
321-334, 2000.
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Flux Balance Model

• Perform mass balances about each species
• Assume measurements of glucose influx J0
  acetaldehyde/pyruvate efflux J are available
• Square matrix solve for unknown fluxes