# 2. Mathematical descriptions. (i) the master equation (ii) Langevin theory. 3. Single cell measurements

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1 1. Why stochastic?. Mathematical descriptions (i) the master equation (ii) Langevin theory 3. Single cell measurements 4. Consequences

2 Any chemical reaction is stochastic. k P d φ dp dt = k d P deterministic equation protein number (P) k/d 0 time (sec) 1000

3 Any chemical reaction is stochastic. k P d φ protein number (P) k/d 0 time (sec) 1000

4 Why are chemical reactions stochastic? f A B C 1. Reactants diffuse to find each other in solution.. They must overcome the energy barrier of the reaction. Both events are randomly affected by thermal fluctuations: collisions with other (solvent) molecules

5 When are chemical reactions significantly stochastic? As a reaction only increases or decreases the number of molecules by one or two, it is only when numbers of molecules are small that random timing of individual reactions will matter. noise = standard deviation mean

6 Noise depends on low numbers. 60 mean mean 10, protein number noise = noise = time (sec) time (sec)

7 Gene expression in bacteria

8 Transcription: DNA to mrna QuickTime and a Cinepak decompressor are needed to see this picture.

9 Translation: mrna to protein

10 Gene expression in bacteria

11 Measuring noise in single cells.

12 Measuring noise Integrate Green Fluorescent Protein OriC Escherichia coli chromosome

13

14 Example: RNAP varies from cell to cell but gene expression is deterministic P [ RNAP] hence η = P P P =

15 Two colour experiment OriC cfp yfp Look at difference between two colours η int ( C Y )

16 Each colour is controlled by identical regulatory sequences. cfp yfp Correlated expression ( Quiet ) Uncorrelated expression ( Noisy ) fluorescence time fluorescence time

17 Example: RNAP varies but gene expression is deterministic No intrinsic noise η = 1 N 1 N ( C Y ) int = 1 C N Y 0

18 Two types of noise: intrinsic and extrinsic Extrinsic Intrinsic Extrinsic variables: same for each gene e.g. number of free RNAPs, cellular environment Intrinsic variables: different for each gene e.g. number of transcribing RNAPs, number of mrnas

19 PHASE CONTRAST IMAGE

20 CFP IMAGE

21 YFP IMAGE

22 MERGED IMAGE

23 Extracting fluorescence values Phase image

24 Automatic Cell Identification Automatic cell identification

25 What results do we expect?

26 Simulation of constitutive gene expression (Gillespie method) protein number η tot protein conc ( M) mrna number η tot η tot time (units of cell cycles)

27 Experimental results YFP intensity steady state limit cycle extrinsic noise intrinsic noise CFP intensity

28 Experimentally then extrinsic average intrinsic average giving η = η + η tot int ext

29 Definitions total noise η tot = C + Y C Y C Y bracket denotes average over all cells intrinsic and extrinsic noise η int = which obey ( C Y ) C Y η = η + η tot int ext η ext = CY C Y C Y

30 Experimentally, two different noise sources can be distinguished. 1.6 M D Normalized Mean YFP Intensity Intrinsic Noise, η int Extrinsic Noise, η ext Normalized Mean CFP Intensity Constraint: η + η = η int ext tot

31 Intrinsic noise can be small Twin (strong) lac-regulated artificial promoters, based on λ P L yfp cfp yfp=cfp ΔlacI Intensity 1.0 η int = ( ) η ext = ( ) η tot = ( )

32 Negative transcriptional control P =Promoter (DNA sequence) R =Repressor (protein) Repressor gene R P R gene x Off On R R R mrna Nothing protein X

33 Intrinsic noise depends on expression level yfp cfp WT (RPR37) + IPTG WT (RPR37) decrease expression 3-fold Intensity = 1 η int = ( ) η ext = ( ) Intensity = 0.03 η int = 0.5 (0.-0.7) η ext = 0.3 ( )

34 1. Gene expression is both intrinsically and extrinsically stochastic.. Extrinsic noise is usually greater than intrinsic noise. 3. Intrinsic noise decreases as gene expression levels increase. Elowitz et al. (00) Swain et al. (00)

35 Red: protein Green spots: mrna scale bar: 1 μm

36 Variance in mrna against mrna numbers Naïve prediction:

37 Time course of mrna numbers: mrna is produced in bursts mrna mrna of sister cell piecewise linear fit time (min) smoothed version of red data

38 More complex model DNA off DNA on TRANSCRIPTION mrna TRANSLATION mrna + protein DEGRADATION DEGRADATION

39 Consequences

40 Noise affects the reliable function of genetic networks, by affecting timing. e.g. biological rhythms (Elowitz & Leibler, Barkai & Leibler) deterministic stochastic

41 Noise can also be exploited by cells, e.g. avoiding the immune response. Salmonella typhimurium from Molecular Cell Biology, 4 th ed

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