Dr' Eduardo Mendoza

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Lecture 6
Steady State Analysis for S-Systems

• Dr. Eduardo Mendoza
• Physics Department
• Mathematics Department Center for
  NanoScience
• University of the Philippines
  Ludwig-Maximilians-University
• Diliman Munich, Germany
• eduardom_at_math.upd.edu.ph
  Eduardo.Mendoza_at_physik.uni-muenchen.de

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Topics to be covered

• 6.0 Review Exercises
• 6.1 Benefits of S-System Representation
• 6.2 Examples
• Linear Pathway with Feedback
• Zero Steady States
• 6.3 Regular S-Systems
• 6.4 Exercises
• 6.5 Irregular S-Systems
• Zero rate constant (no steady state)
• System matrix is singular (infinitely many)
• 6.6 Local Stability
• 6.7 Transition Time
• 6.8 Exercises/Homework

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6.0 Homework

• Exercises 4-5, p. 186

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6.1 Benefits of S-System Representation

• Existence of a steady state
• Kinetic orders
• Explicit solution for the steady state
• Kinetic orders
• Rate constants
• Gains and parameter sensitivities
• Kinetic orders
• Local stability conditions
• Kinetic orders
• Turnover numbers

Savageau, J. Mol. Evolution 5199 (1975)
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Review Steady State

• If all equations are balanced (i.e., production
  is balanced by depletion), then dXi/dt 0, for
  i1,,n.
• Thus the steady-state is achieved at
• 0 ai Õj1nm Xjgij - bi Õj1nm Xjhij
• or
• ai Õj1nm Xjgij bi Õj1nm Xjhij
• A steady state is characterized by the condition
  that no metabolite is changing (i.e., that dXi/dt
  0) and they remain constant..

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Why Steady-state (stability) analysis

• Many important aspects of a system at or close to
  a steady state can be analyzed without having to
  solve the differential equations
• Most biochemical systems in nature operate close
  to a steady state, in which inputs and outputs
  are in balance
• Eg even in a disease state, a metabolic system
  typically is in a steady state

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6.2 Examples

• Steady-state equations are linear systems of
  equations ? solutions via standard linear algebra
  
• Later will handle the general case
• Regular S-Systems
• Irregular S-Systems
• Zero rate constant (no steady state)
• System matrix is singular (infinitely many)

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Linear Pathway with Feedback

• Start with a simple example

X2
X1
X3
T

• S-System equations?
• Which parameters?
• Steady-state solutions?

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Infinite Number of Steady States

• Example 1
• dX1/dt 2X20.75 2 X1,
• dX2/dt 1.2 X1- 1.2X20.5
• Example 2

• PLAS Recycle.plc
• (for Example 2)
• X1' X3 - 0.1 X1 X2.2
• X2' 0.1 X1 X2.2 - 9 X2
• X3' 9 X2 - X3
• Initial values
• X1 60
• X2 30
• X3 10
• Solution parameters
• t0 0
• tf 5
• hr 0.1

X3
X1
X2

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6.3 Regular S-Systems

• General derivation of system of linear equations
• Matrix notation A y b where
• A (aij) is an n x n matrix with aij gij -
  hij ,
• y (yj) is a column vector with yj ln Xj,
• b (bj) is a row vector with bj ln(ßi/ai)
• Regular S-System if A is invertible (A-1 exists

det A ? 0 )
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Regular S-Systems with independent variables

• Derivation of system of linear equations
• Matrix notation AD yD b - AI yI where
• AD (aij) is an n x n matrix with aij gij -
  hij ,
• AI (aij) is an n x m matrix with aij gij -
  hij ,
• yD (yj) is an n-column vector with yj ln Xj,
  
• yI (yj) is an m-column vector with yj ln
  Xj,
• b (bj) is an n-row vector with bj ln(ßi/ai)
• Explicitly yD AD-1b AD-1AIyI

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6.4 Exercises

• Exercises 1.1-1.2, 3.1-3.4
• Discussion (to check understanding of
  steady-state concepts)
• Exercise 7
• Homework
• Exercise 8
• Exercise 9

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6.5 Irregular S-Systems

• Example 1 At least one of the rate constants is
  zero
• Do steady states exist?
• Why or why not?
• Discussion
• a 0 What happens?
• ß 0 What happens?

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Singular S-Systems (1)

• PLAS Cycle.plc
• X1' X20.5 - 2 X1
• X2' 2 X1 - X20.5
• X1 1
• X2 1
• t0 0
• tf 100
• hr 1

• Example 2 A very simple cyclic pathway

det A ?
X2
X1
Steady states?
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Singular S-Systems (2)

• Example 3
• .
• dX1 /dt 2 X1 2 X2
• dX2 /dt X1 2 X2
• dX3 /dt X3 X32

• Graph (map) ?

X2
X1
X3
det A ?
Steady states?
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6.6 Local Stability

• Standard definitions and techniques from
  dynamical systems
• Via Fundamental Theorem of Hartman and Grobman
  under appropriate assumptions, a non-linear
  system can be approximated by a linear system
• Linear system behavior described by tangent
  vector (in turn described by eigenvalues)
• Routh-Hurwitz method for determining positive
  real parts of eigenvalues
• Very useful for
• Small systems, where interest is for a whole
  family
• Larger systems with a very regular and convenient
  structure, eg linear pathways with feedback

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Eigenvalue (of a matrix)

• Let A be an n x n matrix with complex
  coefficients.
• An eigenvalue ? of A is a complex number such
  that there is a non-zero vector v with Av ?v.
  v is called an eigenvector of A
• Eigenvalues of A are given by the characteristic
  equation det(A- ?I) 0.

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Eigenvalues and tangent vectors

• A solution F(x,y,z) is a curve in 3-space
• For small movements (in time), a tangent vector
  is defined
• Eigenvalues describe the speed with which the
  concentrations change (along local axes)

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Stability Analysis (1)

• Routh Hurwitz Method
• Define F-Factors and use to compute
  characteristic polynomial
• Procedure fails if any of the coefficients of
  char polynomial are 0
• Form Routh-Hurwitz array to compute for solutions
• Count number s of sign changes among elements in
  the first column
• If s 0, system is stable, unstable otherwise

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Stability Analysis (2)

• Linear Pathway with Feedback

X2
X1
X3
T

• S-System equations?
• Which parameters?

Voit, Eq 6.64, p. 212
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6.7 Transition time (1)

• Transition time
• average time a molecule remains in a pathway
• Measure for throughput of a system
• Defined here (several others exist) as sum of
  metabolites at steady-state, divided by
  steady-state efflux.
• t S Xi, S / Vn- with i 1,...,n

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Transition time (2)

• Example
• (Longer) linear pathway
• with feedback
• X1' 2 - 2 X10.5 X5-1
• X2' 2 X10.5 X5-1 - 4 X20.5
• X3' 4 X20.5 - 4 X30.8
• X4' 4 X30.8 - 1 X4
• X5' 1 X4 - 4 X5.5

• Initial values
• X1 2
• X2 1
• X3 1
• X4 1
• X5 1
• t0 0
• tf 100
• hr 1

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6.8 Exercises/Homework

• Exercises 8, 9, 14.1,15

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Thanks for your attention !

• Questions?