Mathematical Modeling of the Heat Shock Response in E' coli

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Feedback Regulation of Bacterial Stress
Response Mustafa Khammash In collaboration
with H. El-Samad, Iowa State UniversityH.
Kurata, Kyushu Institute of Tech.John Doyle,
Caltech
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Dependence of E. coli Growth on Temperature
Normal growth
Arrhenius relationship velocity of chemical
reactions and absolute temperature (T) are
related by
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Heat-Shock Response

• High temperatures lead to heat induced stress due
  to a large increase in protein
  unfolding/misfolding
• The heat-shock response is a protective cellular
  response to deal with heat-induced protein
  damage.
• Involves building and dispatching heat-shock
  proteins (HSPs)
• Chaperones e.g. DnaK, DnaJ, GrpE, GroES, GroEL,
  
• Proteases e.g. Lon, FtsH, HslVU, Clp,

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Function of Heat-Shock Proteins
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Heat-Shock Response

• In E. coli, induction of the heat-shock response
  is attributed to an increase in the regulator s32
  
• Upon a temperature upshift from 30C to 42C,
• Induction phase s32 levels rapidly increase
• Adpatation phase s32 levels then gradually
  decrease reaching a new steady-state (higher than
  at 30C)

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Heat-Shock Gene Transcription
DNA
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mRNA Translation
Heat-Shock Proteins
mRNA
ribosomes
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Regulation of HSP Synthesis
Achieved through tight regulation of s32 at 3
levels

• Heat-Induced s32 Synthesis
• Translational induction is due to temperature
  melting of rpoH mRNA secondary structure

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• Regulation of s32 Activity
• Chaperones inhibit s32 activity by sequestering
  s32 away from RNAP. Unfolded proteins increase
  s32 activity.

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• Regulation of s32 Degradation
• s32 is transiently stabilized upon temperature
  upshift

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Regulation of Heat-Shock Response
Transcription
Heat
-
Translation
Degradation
-
Activity
HslVU FtsH
Proteases
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Mechanisms Modeled

• s32 synthesis (temperature dependent)
• Binding of s32 to RNAP
• Binding of s32RNAP complex to gene promoters
• Transcription and translation of chaperones
  (represented by DnaK)
• Transcription and translation of FtsH
• Transcription and translation of other proteases
  (represented by HslVU)

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• Competition among DnaK and RNAP for s32
• Degradation of s32 when unprotected by RNAP
• Protein denaturing (temperature dependent)
• Binding of DnaK to unfolded proteins
• Protein folding via DnaK

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Mathematical Model
Protein Synthesis
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Binding Equations
Mass Balance Equations
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Feedback and feedforward Architecture of the
Heat-Shock Response
Dnak translation transcription dynamics
degradation rate
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Full Model Simulations
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FtsH Null Mutant Simulations
Mutant
Mutant
Wild Type
Wild Type
42o
30o
FtsH Null mutant
Wild Type
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Sensitivity Analysis
Let l be a given system parameter, e.g. binding
constant, translation rate, degradation rate,
etc.
The sensitivity of x(t) and y(t) to variations in
the parameter l is given by
Sensitivity Equations
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Sensitivity to Model Parameters
Sensitivity of DnaK and folded proteins to model
parameters
-1
10
-2
K-s32/dnaK
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K-s32/RNAP
Folded Proteins
K-s32RNAP/D
transcription rate
DnaK
-3
10
Sensitivity
-4
10
-5
10
100
200
300
400
500
600
700
800
Time (min)
42o
30o
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Effect of Sequestration on Sensitivity
Sensitivity of dnaK to model parameters (with and
w/o sequestration loop)
1
10
0
10
No sequestration loop
-1
10
Wild type
-2
Sensitivity of dnaK
10
K-s32/dnaK
K-s32dnaK/D
transcription rate
-3
10
-4
10
-5
10
100
200
300
400
500
600
700
800
Time (min)
42o
30o
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Basic Architecture of HS Response
heat
HSP transcription/ translation protein folding
Binding to promoter
translation
Sequestration
degradation
Sequestration Local loop
Robustness to Parameteric Uncertainty Lowering
the metabolic burden
Degradation Outer loop
Stability and Regulation of
Speed Efficiency of Protein Folding
Feedforward
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Stochastic vs. Deterministic

• Total number of sigma-32 molecules per cell is
  very small (30 per cell)
• Number of free sigma-32 molecules per cell is
  even smaller
• Does it make sense to treat these quantities as
  concentrations?
• Stochastic models need to be considered

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Free
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Current Research Questions

• Understanding the advantages of the present
  control architecture with respect to
• Robustness to variations in certain parameters,
  e.g. rate constants
• Hypersensitivity to other parameters
• Efficiency vs. robustness
• Are there general principles that are used in
  other stress responses?

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Control Theory in Biological Systems

• Feedback regulation mechanisms are ubiquitous
• Bring out the dynamic nature of biochemical
  interactions
• Explain interactions in the context of regulation
• New perspective on biological regulation
• Robustness, adaptation, amplification, isolation,
  and nonlinearity
• Many similarities with engineering systems

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• Identify functional biological modules and any
  constraints on them
• Constraints impose functional requirements
• Easier to understand/predict the function of
  sub-modules

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Acknowledgements

• Carol Gross, UCSF
• Tau-Mu Yi, Caltech