Section 5: Reverse Engineering Biological Systems

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Section 5 Reverse Engineering Biological Systems
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Why is this Section Included in the Tutorial?

• You need good models to perform control analysis.
• Concepts and techniques used to design feedback
  systems are helpful for analyzing feedback
  systems constructed by Nature.
• It is still cold and dark outside.

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Model
Data

• Biochemical
• Time-course
• Dose-response
• Genetic

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How to Fit Model to Data?

• System Identification
• Statistical methods
• Prediction/Machine Learning
• Parameter optimization
• Same basic techniques.

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Fitting a Function to Data Points
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System Identification Loop

• Design experiments and collect data.
• Polish the data.
• Select and define the model structure.
• Compute the best model (estimate parameters)
  based on
• data and a given criterion of fit.
• Examine the model properties.
• If not satisfied, go back to Step 3 and try a new
  model
• structure or go back to Step 4 and try a new
  estimation
• procedure.

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Classic System ID Focuses on Linear Systems

• Simple transfer function representation.
• Linear state-space methods.
• Sometimes can use linear regression to estimate
  parameters.
• Simpler statistical interpretation of parameter
  estimates.

• But biological systems are nonlinear.
• Linearize nonlinear system.
• Generalize these ideas to nonlinear systems

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What Many of Us Do

• Convert arrow diagram into reaction diagram.
• Write ODEs (model) based on mass-action or
  Michaelis-Menten kinetics.
• Guesstimate reasonable starting values for
  parameters (rate constants, total concentrations,
  etc.).
• Collect data.
• Decide on criterion of fit (e.g., least-squares).
• Decide on training/testing protocol (e.g.,
  hand-crafting versus cross-validation).
• Estimate parameters using optimization procedure.
• Evaluate model based on error residual.

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Model Validation in the Presence of Noise and
Uncertainty

• Your model has so many parameters you can fit
  any data.
• My data measurements werent that good.
• Your model is leaving out a lot of things.

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Without Noise, Identification is Easy

• Without noise,
• We know Y and f, and hence can solve for q.
• With noise,
• We dont know x(t) it has to be estimated using
  the Kalman filter.
• The two s will not be the same.

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Statistical Interpretation

• Data y is generated from some probability
  distribution.
• Parameter estimate also forms a distribution.
• Maximum likelihood estimate
• Least squares estimate (LSE) is the MLE for data
  with normally distributed errors.

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Parameter Estimate Distribution

• Bayes Theorem
• Cramer-Rao limit on minimum spread (variance) of
  this distribution
• In this framework, is it possible to invalidate a
  model ( )?

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Alternative Approach Model Invalidation

• Test whether data is inconsistent with model
  structure.
• Define feasible parameter space (empty?).
• Uncertainty in model and data is built-in.
• Framed as convex optimization problem.
• Model is falsifiable.

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Feasible Parameter Space

• Michael Frenklach, Andy Packard and Pete Seiler
• Mechanical Engineering
• University of California
• Berkeley, CA

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Procedure for Invalidating a Model
(1)
 
(2)
(3)
Can we find lk such that B is nonnegative and
thus have a contradiction?
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Sum of Squares Program (SOSP)

• Can we write B as a sum of squares?
• The feasible space is convex convert this SOSP
  into a convex optimization problem (SDP).
• This can be solved using SOSTOOLS in Matlab
  (Prajna, Papachristodoulou, and Parrilo
  http//www.cds.caltech.edu/sostools/)

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Convex Optimization Problem
Feasibility problem
Optimization problem

• For convex optimization problems, any local
  solution is also global.
• Examples include LP, QP, SDP.
• Powerful computational algorithms for solving
  these problems.

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Example yeast pheromone response

• We were able to identify the nonnegative li and
  the corresponding sum of squares expression for
  B(r), thereby invalidating the model.
• We were able to identify the data points
  responsible for the invalidation.
• If the model had been validated, then we could
  define the feasible parameter space and use the
  Model/Data for predictions.

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Summary of Section 5

• Model identification and model invalidation are
  complementary approaches.
• Explicitly describe the uncertainty in the model
  and data.
• Be careful when you say We validated the model
  using the following data.

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Summary of Tutorial

• Robustness is a defining feature of living
  systems.
• An appreciation of control is essential to
  understanding the Logic of Life.
• Always include the feedback loops in your
  diagrams and modeling.
• Collaborate with your neighborhood control
  theorist.

www.cds.caltech.edu/tmy/tutorial.htm