MiniCourse on Mathematical Modeling of Biological Systems

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Mini-Course onMathematical Modeling of
Biological Systems

• Frank Doyle
• Dept. of Chemical Engineering
• University of Delaware
• [phone]
• fdoyle_at_udel.edu

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General Descriptions of Metabolic Systems

• In general, dynamics must be accounted for
• A number of simplifying assumptions invoked
• steady-state
• simple goals or costs functions

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Cybernetic Models

• Introduced by Ramkrishna co-workers
• Key tenet microorganisms implement regulatory
  strategies with outcomes consistent with a
  desired goal
• Requires formulation of objective function for
  specific structural units from which cybernetic
  control variables are defined (enzyme synthesis
  enzyme activity)
• Does not replace detailed modeling of metabolic
  pathways or ignore biochemical details - rather
  it supplies the regulatory mechanisms
• Wide success in describing varying bacterial
  uptake patterns of mixed, substitutable
  substrates in batch, fed-batch, and continuous
  reactors

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Basic Postulates

• The primary goal of a microorganism is to
  maximize growth rate.
• The objective of promoting any irreversible
  linear pathway is the maximization of that
  pathways end product.
• All end products produced at a branch point are
  essential for growth therefore cellular growth
  stops when any one end product is absent.
• The objective of promoting any irreversible
  branched pathway is the maximization of the
  mathematical product of the end products produced
  by one or more branch points.
• The objective of promoting any irreversible
  cyclic pathway is the maximization of the
  mathematical product of all the intermediates of
  the cycle whether or not they were withdrawn for
  further utilization in order to maintain the
  integrity of the cycle.

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(contd)

• In the presence of competing processes, whether
  substitutable or complementary in nature, the
  enzyme catalyzing the process that provides for
  the greatest return will be activated to the
  greatest extent. The appropriate measure of the
  returns received from promoting a given process
  is dependent upon the process type, e.g.,
  substitutable or complementary.
• Metabolic regulation is present in a hierarchical
  structure compromising both local and global
  levels
• As the status of a substrate is altered form
  limiting to nonlimiting, the excess substrate is
  available for the synthesis of additional
  structural and energetic resources.

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Cybernetic Framework
BIOMASS
ENZYMES
INTERNAL RESOURCES INTERMEDIATES
C
N
ENZYME SYNTHESIS
REGULATORY PROCESSES
S
P
METABOLISM (INTERMEDIATE SYNTHESIS)
etc.
O
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Optimal Control vs. Instantaneous Optimization

• Endpoint optimization
• ui fraction critical resources
• Instantaneous optimization
• ui maximizes return

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Example Klebsiella pneumoniae

• glucose feed, watch cell density as output
• Model equations
• cell density
• growth enzyme
• maintenance enzyme
• cellular resource level
• substrate concentration

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Key Rate Functions

• growth rate
• growth enzyme production rate
• maintenance enzyme production rate
• rate of resource synthesis

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PHB production in Alcaligenes eutrophus

• Motivation
• polymer biodegradable, thermoplastic
• process green synthesis route for polyesters
• Potential applications
• medical (implants, drug delivery)
• film, packaging
• Cybernetic model developed by Varner Bailey

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Pathway Structure

• p3 PHB

storage pathway model
cr
example cybernetic variable governing synthesis
of key biomass growth enzyme, eg
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Cybernetic Variable Governing Synthesis of Key
Growth Enzyme
Cybernetic variable adjusting synthesis rate of eg
Ammonium nitrate concentration in reactor
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Model Equations
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Parameter Identification

• Model parameters identified from batch cultures

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Continuous Culture with Biomass Recycle
biomass recycle
dilution rate (glucose and ammonium feed)
PHB productivity
ideal separator (microfiltration)
reactor
glucose measurement
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Wash-out Disturbance
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Cybernetic Variable Profiles

• Cybernetic variables governing synthesis of
  enzymes for PHB storage and biomass growth

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Cybernetic Variables Governing Enzyme Activity

• Cybernetic variables are non-smooth functions
• cybernetic variable modulating activity of
  storage enzyme
• Batch culture profile for v3
• Gradient based optimization?

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Generalized Framework (Varner, 2000)
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