Assigning Numbers to the Arrows

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Assigning Numbers to the Arrows

• Parameterizing a Gene Regulation Network by using
  Accurate Expression Kinetics

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Overview

• Motivation
• Gene Regulation Networks Background
• Our Goal
• Our Example
• Parameterizing Algorithm
• Results

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Motivation

• Understand regulation factors for different genes
• Can help understand a genes function
• If we can understand how it all works we can use
  it for medical purposes like fixing and
  preventing DNA damage!

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Background Gene Regulation Networks(1)

• Dynamically orchestrate the level of expression
  for each gene
• How? Control whether and how vigorously that gene
  will be transcribed into RNA (biological stuff)

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Background Gene Regulation Networks(2)

• Contains
• 1. Input Signals environmental cues,
  intracellular signals
• 2. Regulatory Proteins
• 3. Target Genes

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Our Goal

• Assign parameters to a Gene Regulation Network
  based on experiments
• - production of unrepressed promoter. the
  maximum production
• - concentration of repressor at half maximal
  repression. The bigger it is the earlier the
  earlier the gene becomes active and the later it
  becomes inactive again

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Our Example(1)

• Escheria coli bacterium
• SOS DNA repair system used to repair damage
  done by UV light
• 8 (out of about 30) gene groups (operons)

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Our Example(2)

• Simple network architecture recall what we saw
  last week SIM (Single Input Module)
• All genes are under negative control of a single
  repressor (a protein that reduces gene levels)

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Parametrization Algorithm
Definitions
- the activity of promoter i in experiment j as
function of time
- effective repressor concentration in
experiment j as function of time
- production rate of the unrepressed promoter i
- k parameter of promoter i
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Parametrization Algorithm 1Trial Function
Why? Michaelis-Menten form a very useful
equation in modeling biological behavior.
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Parametrization Algorithm 2Data Preprocessing(1)

• Smoothing the signals using a hybrid
  Gaussian-median filter with a window size of five
  measurements
• Five time points are taken, sorted and the
  average of central three points is taken to be
  the signal.

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Parametrization Algorithm 2Data Preprocessing(2)
Some more definitions
- the activity of promoter i as a function of
time
- GFP fluorescence from the corresponding
reporter as a function of time
- corresponding Optical Density as a function of
time
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Parametrization Algorithm 2Data Preprocessing(3)

• The signal is smooth enough to be differentiated
• The activity of promoter i is proportional to the
  number of GFP molecules produced per unit time
  per cell

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Parametrization Algorithm 2Data Preprocessing(4)

• The activity signal is smoothed by a polynomial
  fit of sixth order to

• The smoothing procedure captures the dynamics
  well, while removing noise
• Data for all experiments is concatenated and
  normalized by the maximal activity for each
  operon

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Parametrization Algorithm 3Parameter
Determination(1)

• To determine parameters in equation 1 based on
  experimental data we transform it into a bilinear
  form

where
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Parametrization Algorithm 3Parameter
Determination(2)

• Now, the matrix

where N is for genes and M for time points, is
modeled by two vectors of size N
and one vector of size M

• 2NM variables

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Parametrization Algorithm 3Parameter
Determination(3) some algebra

• The standard method of least mean squares
  solution for such a problem
  uses SVD (Singular Value Decomposition)
• The mean over i of

is removed
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Parametrization Algorithm 3Parameter
Determination(4) some algebra

• A(t) is the SVD eigenvector with the largest
  eigenvalue of the matrix
• This is the covariance matrix

• Results for A(t) are normalized to fit the
  constraints
• Alternative normalization add points with A0
  and

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Parametrization Algorithm 3Parameter
Determination(5) some algebra

• Perform a second round of optimization for by
  using a nonlinear least mean squares solver to
  minimize

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Parametrization Algorithm 4Error Evaluation(1)

• The mean error for promoter i is given by

• where T is the total time of the experiment
• This is considered the quality of the data
  model in describing the data

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Parametrization Algorithm 4Error Evaluation(2)

• The error estimate for the parameters is
  determined by using a graphic method

is plotted vs. A(t)
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Parametrization Algorithm 4Error Evaluation(3)

• From maximal and minimal slopes of the graphs
  the error for is determined

• From maximal and minimal intersections with the
  y axis the error for is determined

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Parametrization Algorithm 5Additional Trial
Function(1)

• An extension of the model to the case of
  cooperative binding a regulator can be a
  repressor for some genes and an activator for
  others, and with different measures

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Parametrization Algorithm 5Additional Trial
Function(2)

• Hill coefficient for operon i

Hill coefficient? A coefficient that describes
binding
- repression
- activation
- no cooperation
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Parametrization Algorithm 5Additional Trial
Function(3)
Our example good comparison between measured
results and those calculated with trial
function suggest there may be no significant
cooperativity in the repressor action
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Results Promoter Activity Profiles(1)

• After about half a cell cycle the promoter
  activities begin to decrease
• Corresponds to the repair of damaged DNA

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Results Promoter Activity Profiles(2)

• The mean error between repeat experiments
  performed of different days is about 10

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ResultsAssigning Effective Kinetic Parameters

• The error is under 25 for most promoters

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ResultsDetection of Promoters with Additional
Regulation

• Relatively large error may help to detect operons
  that have additional regulation.
• Examples
• 1. lacZ very large error (150)
• 2. uvrY recently found to participate in
  another system and to be regulated by other
  transcription factors (45 error)

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ResultsDetermining Dynamics of an Entire System
Based on a Single Representative(1)

• Once the parameters are determined for each
  operon, we need to measure only the dynamics of
  one promoter in a new experiment to estimate all
  other SOS promoter kinetics

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ResultsDetermining Dynamics of an Entire System
Based on a Single Representative(2)

• The estimated kinetics using data from only one
  of the operons agree quite well with the measured
  kinetics for all operons
• Same level of agreement found by using different
  operons as the base operon

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ResultsDetermining Dynamics of an Entire System
Based on a Single Representative(3)
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ResultsRepressor Protein Concentration Profile

• Current measurements dont directly measure the
  concentration of the proteins produced by these
  operons, only the rate at which the corresponding
  mRNAs are produced
• The parameterization algorithm allows calculation
  of the transcriptional repressor - A(t), directly.

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Summary

• We can apply the current method to any SIM motif,
  in gene regulation networks
• The method wont work with multiple regulatory
  factors

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Questions?

• Thank You For Listening!