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

2. Overview Motivation Gene Regulation Networks Background Our Goal Our Example Parameterizing Algorithm Results

3. 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!

4. 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)

5. Background: Gene Regulation Networks(2) Contains: 1. Input Signals: environmental cues, intracellular signals 2. Regulatory Proteins 3. Target Genes

6. 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

7. 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)

8. 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)

9. 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

10. Parametrization Algorithm 1: Trial Function Why? Michaelis-Menten form: a very useful equation in modeling biological behavior.

11. 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.

12. Parametrization Algorithm 2: Data Preprocessing(2) - 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 Some more definitions:

13. 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

14. Parametrization Algorithm 2: Data 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

15. Parametrization Algorithm 3: Parameter Determination(1) To determine parameters in equation [1] based on experimental data we transform it into a bilinear form: where:

16. Parametrization Algorithm 3: Parameter 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: 2N*M variables

17. 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 is removed:

18. Parametrization Algorithm 3: Parameter 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 A=0 and

19. Parametrization Algorithm 3: Parameter Determination(5) – some algebra Perform a second round of optimization for by using a nonlinear least mean squares solver to minimize

20. Parametrization Algorithm 4: Error 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

21. Parametrization Algorithm 4: Error Evaluation(2) The error estimate for the parameters is determined by using a graphic method: is plotted vs. A(t)

22. Parametrization Algorithm 4: Error 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

23. 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:

24. Parametrization Algorithm 5: Additional Trial Function(2) -Hill coefficient for operon i Hill coefficient? A coefficient that describes binding - repression - activation - no cooperation

25. Parametrization Algorithm 5: Additional 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

26. Results: Promoter Activity Profiles(1) After about half a cell cycle the promoter activities begin to decrease Corresponds to the repair of damaged DNA

27. Results: Promoter Activity Profiles(2) The mean error between repeat experiments performed of different days is about 10%

28. Results: Assigning Effective Kinetic Parameters The error is under 25% for most promoters

29. 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)

30. 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

31. 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

32. Results: Determining Dynamics of an Entire System Based on a Single Representative(3)

33. 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.

34. Summary We can apply the current method to any SIM motif, in gene regulation networks The method won’t work with multiple regulatory factors

35. Questions? Thank You For Listening!