Systems Biology and Numerical Analysis

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Systems Biology and Numerical Analysis

• Stephanie Taylor
• Ph.D. Candidate
• Department of Computer Science, UCSB
• March 1, 2006

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We are LivingIn a Bacterial World
Staphylococcal Enterotoxin B (SEB)
(CDC MMWR 1983)
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We are LivingIn a Bacterial World
Staphylococcal Enterotoxin B (SEB)
(CDC MMWR 1983)
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SEB
SEB is cleared from the system by the kidneys
But when there is too much SEB, the kidney cells
die (apoptosis).
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SEB-Induced Apoptosis

• How does a kidney cell react to SEB?
• Can we stop it from killing the cell?

Truth in Advertising These questions will not
be answered during this talk. This is an active
area of research!
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Understanding Biological Systems
Systems
Biology

• Capture interactions between components
• Quantify dynamics
• Elucidate more complicated mechanisms

• Determine components of system
• Perform experiments
• Reveal certain mechanisms

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Process

• Experiment
• Explain with Cartoon
• Translate Cartoon Interactions into Mathematical
  Expressions
• Combine Expressions into an ODE Model
• Analyze the Model

(Fall, Marland, Wagner, Tyson, Computational
Cell Biology, 2002)
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SEB Experimentation
cluster analysis
gene arrays
(Zhang, Bio-SPICE Technical Report, 2005), (TJU,
Bio-SPICE Use Case Reports, 2005)
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Process

• Experiment
• Explain with Cartoon
• Translate Cartoon Interactions into Mathematical
  Expressions
• Combine Expressions into an ODE Model
• Analyze the Model

(Fall, Marland, Wagner, Tyson, Computational
Cell Biology, 2002)
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SEB-Induced Apoptosis Cartoon
(Jason Shoemaker,2005)
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Process

• Experiment
• Explain with Cartoon
• Translate Cartoon Interactions into Mathematical
  Expressions
• Combine Expressions into an ODE Model
• Analyze the Model

(Fall, Marland, Wagner, Tyson, Computational
Cell Biology, 2002)
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SEB-Induced Apoptosis Model

• Each node represents an entity (such as a
  protein)
• Each edge represents a reactions
• We construct one ODE for each node.

r14_k
r21_k
(Jason Shoemaker,2005)
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SEB-Induced Apoptosis Model
parameter
r14_k
r21_k
state
(Jason Shoemaker,2005)
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Process

• Experiment
• Explain with Cartoon
• Translate Cartoon Interactions into Mathematical
  Expressions
• Combine Expressions into an ODE Model
• Analyze the Model

(Fall, Marland, Wagner, Tyson, Computational
Cell Biology, 2002)
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SEB-Induced Apoptosis Model


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SEB-Induced Apoptosis Model

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SEB-Induced Apoptosis Model

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Process

• Experiment
• Explain with Cartoon
• Translate Cartoon Interactions into Mathematical
  Expressions
• Combine Expressions into an ODE Model
• Analyze the Model

(Fall, Marland, Wagner, Tyson, Computational
Cell Biology, 2002)
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Simulating the Model

• How do we integrate the ODEs over time to see
  the dynamics?
• If we know what the protein concentrations are at
  time 0, we can use the rate information to
  determine the concentrations at later times.
• An analytical solution is too difficult (or
  impossible) to find.
• So, we solve it numerically

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Forward (Explicit) Euler

• First step Discretize Time

y
time
t1
tn
h
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Forward (Explicit) Euler

• Taylor Series Expansion

X
If the timestep h is small, then this term will
be close to 0.
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Forward (Explicit) Euler

• Taylor Series Expansion
• Forward Euler Method

X
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Forward (Explicit) Euler
y
time
t1
tn
h
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Forward (Explicit) Euler
y
time
t1
tn
h
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Forward (Explicit) Euler
y
time
t1
tn
h
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Forward (Explicit) Euler

• If the true solution behaves well, and if our
  timestep h is small enough, then Forward Euler
  will give us a reasonable result.
• However, it is inefficient and has only first
  order precision in h.
• Taylor Expansion
• Forward Euler

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How to get Fancier

• Variable timestep sizes
• Higher Order Methods
• Multi-Step Methods
• Runge-Kutta Methods
• Adams-Moulton Methods
• Backward Differentiation Formula (BDF)
• Software for solving ODEs
• XPP
• DASPK

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Simulation Results
BAD
state indicating apoptosis
state indicating apoptosis
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Simulation Results

• Didnt have expected dynamics
• Cell died whether or not SEB was present
• What is wrong with the model?
• Either our kinetics are incorrect
• Or we arent capturing enough of the players
• We are pretty sure the kinetics are correct. So
  how do we choose where to expand the model?
• Use human intuition
• Is there a way the computer can help? With 77
  states, we have lots of information.

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Refining the SEB Model

• What if we knew what parts of the model were
  having the strongest effect on the behavior?
  Where are the hotspots?
• We can find the hotspots using sensitivity
  analysis

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Sensitivity Analysis

• What effect does a small perturbation in a
  parameter have upon the state of the system at
  time t?

p0.2Dp Dp 0.05
p0.2
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Sensitivity Analysis

• Solve the sensitivity equations along with the
  original system

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Sensitivity Analysis
NT
NY
NP
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Fisher Information Matrix
NP
Weighted Norms of the Sensitivities
NP
S1T
S1
V1
F1
S2T
V2
S2
F2
S3T
V3
S3
F3
S4T
V4
S4
F4
X
X
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Fisher Information Matrix
NP
NP
Overall Parameter Sensitivities
NP
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Fisher Information Matrix
Overall Parameter Sensitivities
NP
323 0.001 0.023 3E5 3E6
3.245 748 0.547 23E3 276 10
The system is highly sensitive to parameters P4,
P5, and P9!
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Hotspots

• So, we did the sensitivity analysis, identifying
  some parts of the pathway that had a large effect
  upon the dynamics of the system
• Did a literature and database search for
  genes/proteins related to the ones in the
  hotspots.
• Expanded the model

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Model Refinement
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Model Refinement
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Model Refinement
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Simulation
PI3K_A
PI3K_A
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Conclusion

• Process
• Experiment
• Explain with Cartoon
• Translate Cartoon Interactions into Mathematical
  Expressions
• Combine Expressions into an ODE Model
• Analyze the Model (Simulation, Sensitivity
  Analysis)
• REPEAT
• And now, for a peek into my software

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BioSens
SBML
FIM
Model dx/dt f(x,p,t)
BioSens
Simulation Sensitivity Computation
Sensitivity Ranking
x(t,p) Si,j
Measurement Selection
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Software Dependencies
BioSens 2.0
XPP
Matlab
libSBML
DASPK 3.0
Xerces
Tapenade
g77, gcc make cygpath
Cygwin
Java VM
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Acknowledgements

• Dr. Rudi Gunawan
• Dr. Tingting Zhang
• Jason Shoemaker
• Dr. Francis J. Doyle, III
• Dr. Linda Petzold

www.cse.ucsb.edu
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Thank You!
Questions?
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Fisher Information Matrix

• FIM represents the amount of information
  contained in data.
• Rankings Diagonal entries represent the effect
  each parameter has on the overall system.

( states)
( parameters)
( timesteps)
( parameters)
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Sensitivity Analysis
NT
NT
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Sensitivity Analysis
SiT
Si
X
X
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Sensitivity Analysis
NP
NP
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Sensitivity Analysis
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Sensitivity Analysis

• Solve the sensitivity equations along with the
  original system