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Assigning Numbers to the
Arrows
Parameterizing a Gene Regulation
Network by using Accurate
Expression Kinetics
Overview
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Motivation
Gene Regulation Networks Background
Our Goal
Our Example
Parameterizing Algorithm
Results
Motivation
• Understand regulation factors for different
genes
• Can help understand a gene’s function
• If we can understand how it all works we can
use it for medical purposes like fixing and
preventing DNA damage!
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)
Background: Gene Regulation
Networks(2)
• Contains:
1. Input Signals: environmental cues,
intracellular signals
2. Regulatory Proteins
3. Target Genes
Our Goal
• Assign parameters to a Gene Regulation
Network based on experiments:
 - production of unrepressed promoter. the
maximum production
k- 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
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)
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)
A
X1
X2
...
Xn
Parametrization Algorithm
Definitions:
X ij (t ) - the activity of promoter i in experiment
j as function of time
A j (t ) - effective repressor concentration in
experiment j as function of time
 i - production rate of the unrepressed
promoter i
k i - k parameter of promoter i
Parametrization Algorithm 1:
Trial Function
[1] : X ij (t ) 
i
(1  Aj (t ) / ki )
Why?
Michaelis-Menten form: a very useful equation
in modeling biological behavior.
Parametrization Algorithm 2:
Data 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.
Parametrization Algorithm 2:
Data Preprocessing(2)
Some more definitions:
X i (t ) - the activity of promoter i as a function
of time
Gi (t ) - GFP fluorescence from the
corresponding reporter as a function of
time
ODi (t ) - corresponding Optical Density as a
function of time
Parametrization Algorithm 2:
Data 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
X i (t )  [dGi (t ) / dt ] / ODi (t )
Parametrization Algorithm 2:
Data Preprocessing(4)
• The activity signal is smoothed by a
polynomial fit of sixth order to:
log[ X i (t )]
• 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
Parametrization Algorithm 3:
Parameter Determination(1)
• To determine parameters in equation [1] based
on experimental data we transform it into a
bilinear form:
1
 ui (t )  ai  A(t )  bi
X i (t )
where:
1
ai 
 i ki
bi 
1
i
Parametrization Algorithm 3:
Parameter Determination(2)
• Now, the matrix X i (t ) N M
where N is for genes and M for time points, is
modeled by two vectors of size N: ai , bi
and one vector of size M: A(t )
• 2N*M variables
Parametrization Algorithm 3:
Parameter 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 ui (t ) is removed:
ui (t )  ui (t )  mean(ui (t ))
Parametrization Algorithm 3:
Parameter Determination(4) – some
algebra
• A(t) is the SVD eigenvector with the largest
eigenvalue of the matrix: J (t , t ' )   ui (t )  ui (t ' )
i
This is the covariance matrix
• Results for A(t) are normalized to fit the
constraints: A(t  0)  1, min( A(t ))  0
• Alternative normalization: add points with
A=0 and X i  
Parametrization Algorithm 3:
Parameter Determination(5) – some
algebra
• Perform a second round of optimization for
 i , ki by using a nonlinear least mean
squares solver to minimize ( X measured  X predicted )2
Parametrization Algorithm 4:
Error Evaluation(1)
• The mean error for promoter i is given by:
T
1
Ei  
T t 1
X itmeasured  X itpredicted
X itmeasured
where T is the total time of the experiment
• This is considered the quality of the data
model in describing the data
Parametrization Algorithm 4:
Error Evaluation(2)
• The error estimate for the parameters is determined
by using a graphic method:
1
A(t ) 1
 ai  A(t )  bi 

X i (t )
 i ki  i
is plotted vs. A(t)
Parametrization Algorithm 4:
Error Evaluation(3)
• From maximal and minimal slopes of the
1
graphs the error for ai 
is determined
 i ki
• From maximal and minimal intersections with
1
the y axis the error for
is determined
i
Parametrization Algorithm 5:
Additional 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:
X ij (t ) 
i
1  ( Aj (t ) / ki ) H i
Parametrization Algorithm 5:
Additional Trial Function(2)
H i -Hill coefficient for operon i
Hill coefficient? A coefficient that
describes binding
H i  0 - repression
H i  0 - activation
H i  1 - no cooperation
Parametrization Algorithm 5:
Additional Trial Function(3)
Our example: good comparison between
measured results and those calculated with H i  1
trial function suggest there may be no
significant cooperativity in the repressor
action
Results:
Promoter Activity Profiles(1)
• After about half a cell cycle the promoter
activities begin to decrease
• Corresponds to the repair of damaged DNA
Results:
Promoter Activity Profiles(2)
• The mean error between repeat experiments
performed of different days is about 10%
Results:
Assigning Effective Kinetic
Parameters
• The error is under 25% for most promoters
Results:
Detection 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)
Results:
Determining 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
X m (t ) 
1
m
kn  n
(
k m X n (t )
 1)
Results:
Determining 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
Results:
Determining Dynamics of an Entire
System Based on a Single
Representative(3)
Results:
Repressor Protein Concentration
Profile
• Current measurements don’t directly measure
the concentration of the proteins produced by
these operons, only the rate at which the
corresponding mRNA’s are produced
• The parameterization algorithm allows
calculation of the transcriptional repressor A(t), directly.
Summary
• We can apply the current method to any SIM
motif, in gene regulation networks
• The method won’t work with multiple
regulatory factors
Questions?
Thank You For Listening!