Assigning Numbers to the Arrows Parameterizing a Gene Regulation Network by using Accurate...

Assigning Numbers to the Arrows

Parameterizing a Gene Regulation Network by using Accurate

Expression Kinetics

Overview

• 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 Proteins3. Target Genes

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

k

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)

...1X 2X nX

A

Parametrization Algorithm

)(tX ij

Definitions: - the activity of promoter i in experiment j as function of time

)(tAj - effective repressor concentration in experiment j as function of time

i - production rate of the unrepressed promoter i

ik - k parameter of promoter i

Parametrization Algorithm 1:Trial Function

)/)(1()(:]1[

ij

iij ktAtX

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.


)(tX i - the activity of promoter i as a function of time

)(tGi - GFP fluorescence from the corresponding reporter as a function of time

)(tODi - corresponding Optical Density as a function of time

Some more definitions:


• 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

)(/]/)([)( tODdttdGtX iii


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

)](log[ tX i

• 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:

iiii

btAatutX

)()()(

1

where:

iii ka

1

i

ib 1

Parametrization Algorithm 3:Parameter Determination(2)

• Now, the matrix MNi tX )(

where N is for genes and M for time points, is modeled by two vectors of size N: ii ba ,

and one vector of size M: )(tA

• 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 )(tui is removed:))(()()( tumeantutu iii


algebra• A(t) is the SVD eigenvector with the largest eigenvalue of the matrix:

This is the covariance matrix

i

ii tututtJ )'()()',(

• Results for A(t) are normalized to fit the constraints:

• Alternative normalization: add points with A=0 and

0))(min(,1)0( tAtA

iX


algebra• Perform a second round of optimization for

by using a nonlinear least mean squares solver to minimize

ii k,2)( predictedmeasured XX

Parametrization Algorithm 4:Error Evaluation(1)

• The mean error for promoter i is given by:

T

tmeasuredit

predictedit

measuredit

i X

XX

TE

1

1

where T is the total time of the experiment

• This is considered the quality of the data model in describing the data


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

iiiii

i k

tAbtAa

tX 1)(

)()(

1

is plotted vs. A(t)


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

iii ka

1

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

i1

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:

iHij

iij ktAtX

)/)((1)(


0iH

-Hill coefficient for operon i

Hill coefficient? A coefficient that describes binding

- repression0iH - activation

iH

1iH - no cooperation


Our example: good comparison between measured results and those calculated with trial function suggest there may be no significant cooperativity in the repressor action

1iH

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

)1)(

(1)(

tXk

ktX

n

n

m

n

mm



• 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: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!

Assigning Numbers to the Arrows Parameterizing a Gene Regulation Network by using Accurate...

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