Stochastic Gene Expression in Prokaryotes: A Point Process Approach

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1 Stochastic Gene Expression in Prokaryotes: A Point Process Approach Emanuele LEONCINI INRIA Rocquencourt - INRA Jouy-en-Josas Mathematical Modeling in Cell Biology March 27 th 2013 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 1 / 26

2 Introduction Central role of protein production in prokaryotes Central role of protein production Proteins are the core of biologic processes: enzymes, DNA replication machinery,... 50% of the bacteria dry weight 3.5 millions of proteins in each cell 2000 types of proteins produced at any time at any growth condition (volume growth) proteins ranging from few dozens up to 10 5 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 2 / 26

3 Introduction Central role of protein production in prokaryotes Central role of protein production Proteins are the core of biologic processes: enzymes, DNA replication machinery,... 50% of the bacteria dry weight 3.5 millions of proteins in each cell 2000 types of proteins produced at any time at any growth condition (volume growth) proteins ranging from few dozens up to 10 5 A highly consuming process: at each generation, bacteria has to duplicate all proteins more than 85% of cell resources Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 2 / 26

4 Introduction Stochasticity and living organisms Stochasticity in protein production: experimental viewpoint ADk cytoplasm protein 1 1 Yuichi Taniguchi et al. Science (2010), pp Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 3 / 26

5 Introduction Stochasticity and living organisms Stochasticity in bacteria Sources of stochasticity: bacterial cytoplasm: disordered medium main cellular motility mechanism: diffusion in a stiff medium most cellular processes require the encounter of macromolecules (Poisson process) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 4 / 26

6 Introduction Stochasticity and living organisms Stochasticity in bacteria Sources of stochasticity: bacterial cytoplasm: disordered medium main cellular motility mechanism: diffusion in a stiff medium most cellular processes require the encounter of macromolecules (Poisson process) Protein production: inherently stochastic process. Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 4 / 26

7 Model Model Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 5 / 26

8 Model Central Dogma of molecular biology DNA mrna protein TRANSCRIPTION TRANSLATION Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 6 / 26

9 Model 3-stage model: activation λ + 1 λ 1 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 7 / 26

10 Model 3-stage model: transcription λ 2 λ + 1 λ 1 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 7 / 26

11 Model 3-stage model: translation λ 2 λ 3 λ + 1 λ 1 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 7 / 26

12 Model 3-stage model 2 λ 2 λ 3 λ + 1 λ 1 µ 2 µ3 2 Johan Paulsson. Physics of life reviews (2005), pp Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 7 / 26

13 Model 3-stage model Y (t) {0, 1} M(t) N P(t) N λ 2 λ 3 µ 2 µ 3 λ 2 λ 3 λ + Goal: Characterize 1 λ mean and variance of the number of proteins 1P at equilibrium µ 2 µ3 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 8 / 26

14 Model Classic models: Paulsson s survey 3 Properties Assumption: each step has exponentially distributed duration Markovian description of the protein production 3 Johan Paulsson. Physics of life reviews (2005), pp Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 9 / 26

15 Model Classic models: Paulsson s survey 3 Properties Assumption: each step has exponentially distributed duration Markovian description of the protein production Tools Markov processes Fokker-Plank equations explicit analytic formulas of mean and variance as function of the main parameters 3 Johan Paulsson. Physics of life reviews (2005), pp Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 9 / 26

16 Model Classic models: Paulsson s survey 3 Properties Assumption: each step has exponentially distributed duration Markovian description of the protein production Tools Markov processes Fokker-Plank equations explicit analytic formulas of mean and variance as function of the main parameters biologists classically use models because of the explicit characterization of protein fluctuations 3 Johan Paulsson. Physics of life reviews (2005), pp Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 9 / 26

17 Model Classic approach: Exponential assumption: Not each described process has an exponentially distributed duration. Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 10 / 26

18 Model Classic approach: Exponential assumption: Not each described process has an exponentially distributed duration. The duration of the following processes is not exponential protein elongation mrna elongation protein degradation Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 10 / 26

19 Model Protein chain elongation 50S mrna 30S Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 11 / 26

20 Model Protein chain elongation trna fmet trna (charged) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 11 / 26

21 Model Protein chain elongation trna (uncharged) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 11 / 26

22 Model Protein chain elongation Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 11 / 26

23 Model Protein chain elongation Elongation: exponentially distributed elementary steps the sum of exponential random variables is not exponential Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 11 / 26

24 Model Protein chain elongation Elongation: exponentially distributed elementary steps the sum of exponential random variables is not exponential Large number of identical steps (N 400 a.a.) elongation time described as normal random variable Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 11 / 26

25 Model Classic approach: a not completely satisfying description The duration of the following processes is not exponential protein elongation mrna elongation protein degradation New description Gene expression with non-exponential steps requires a new approach. Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 12 / 26

26 MPPP Marked Poisson Point Process (MPPP): new description of gene expression Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 13 / 26

27 MPPP MPPP: mathematical tools & application General distributions: introducing MPPP λ 2 F 2 (dt) Assumptions: mrna births (s n ) follow a Poisson process of parameter λ 2 mrna lifetimes (σ 2,n ) have distribution F 2 (dt) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 14 / 26

28 MPPP MPPP: mathematical tools & application mrna lifetime birth Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 15 / 26

29 MPPP MPPP: mathematical tools & application mrna lifetime σ 2,2 σ 2,3 σ 2,1 E 1 E 2 E 3 birth E i exponential random variables of parameter λ 2. Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 15 / 26

30 MPPP MPPP: mathematical tools & application mrna How many mrnas at time t? lifetime t birth Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 15 / 26

31 MPPP MPPP: mathematical tools & application mrna How many mrnas at time t? lifetime birth s 1 + σ 2,1 t s 3 + σ 2,3 s 2 + σ 2,2 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 15 / 26

32 MPPP MPPP: mathematical tools & application mrna How many mrnas at time t? lifetime y = s + t birth s 1 + σ 2,1 t s 3 + σ 2,3 s 2 + σ 2,2 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 15 / 26

33 MPPP MPPP: mathematical tools & application mrna: general results mrnas at equilibrium: M = 1 {u 0 u+v} N λ2 (du, dv) R R + Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 16 / 26

34 MPPP MPPP: mathematical tools & application mrna: general results mrnas at equilibrium: M = 1 {u 0 u+v} N λ2 (du, dv) R R + Proposition E [M] = δ + λ 2 E [σ 2 ] var(m) = E [M] + 2λ 2 2δ + (1 δ + ) u e Λv (1 F 2 (u))(1 F 2 (u + v)) du dv where F 2 (x) = F 2 ([0, x]), Λ = λ λ 1 and δ + = λ + 1 /Λ Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 16 / 26

35 MPPP MPPP: mathematical tools & application Proteins: general results Proteins at equilibrium: [ ] P = N λ2 (du, dv) R R + 1 {u x u+v} 1 {x 0 x+y} Nλ u 3 (dx, dy) R R + Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 17 / 26

36 MPPP MPPP: mathematical tools & application Proteins: general results Proteins at equilibrium: [ ] P = N λ2 (du, dv) R R + 1 {u x u+v} 1 {x 0 x+y} Nλ u 3 (dx, dy) R R + Proposition E [P] = δ + λ 2 λ 3 E [σ 2 ] E [σ 3 ] var(p) = E [P] + λ 2 λ 2 3δ + + λ 2 2λ 2 3δ + (1 δ + ) R 2 + R 4 + ( ) ( s+t) 0 F 3 (u) du F 2 (t) dt ds s e Λ u 1 u 2 +v 1 v 2 2 F 2 (u i )F 3 (v i ) du i dv i i=1 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 17 / 26

37 MPPP Applications & Model Extensions MPPP Applications & Model Extensions Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 18 / 26

38 MPPP Applications & Model Extensions MPPP 3-Stage Model Application: MPPP 3-Stage Model Y (t) {0, 1} M(t) N P(t) N λ 2 λ 3 λ + 1 λ 1 F 2 (dv) F 3 (dy) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 19 / 26

39 MPPP Applications & Model Extensions MPPP 3-Stage Model Application: MPPP 3-Stage Model Y (t) {0, 1} M(t) N P(t) N λ 2 λ 3 λ + 1 λ 1 µ 2 F 3 (dy) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 19 / 26

40 MPPP Applications & Model Extensions MPPP 3-Stage Model Application: MPPP 3-Stage Model Y (t) {0, 1} M(t) N P(t) N Choices for F 3 (dy) λ + 1 Exponential Normal Deterministic λ 1 explicit close formula depending on model parameters 2 λ 3 λ analytic formula explicit close formula depending on model 2 µ parameters (limit case) F 3 (dy) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 19 / 26

41 MPPP Applications & Model Extensions MPPP 3-Stage Model Deterministic vs Exponential Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 20 / 26

42 MPPP Applications & Model Extensions MPPP 3-Stage Model Application: MPPP 3-Stage Model 3-Stage model with deterministic elongation [ var D (P) = E(P) λ ( 3 1 µ 3 µ 2 + 2λ 2λ 3 (1 δ + )µ 2 Λ 2 µ 2 2 ) (1 e µ 2/µ 3 ) µ 2 µ 3 µ 3 Λ 2 ( µ3 Λ 2 [ 1 e Λ/µ 3 ] [1 e µ 2/µ 3 ] + Λ [ 1 µ Λ 2 ])] 3-Stage model with exponential elongation ( var E (P) = E(P) 1 + λ 3 + λ ) 2λ 3 (1 δ + )(Λ + µ 2 + µ 3 ) µ 2 + µ 3 (µ 2 + µ 3 )(Λ + µ 2 )(Λ + µ 3 ) where Λ = λ λ 1, δ + = λ + 1 /Λ Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 21 / 26

43 MPPP Applications & Model Extensions MPPP 3-Stage Model Conclusions (MPPP) appropriate mathematical tool to describe gene expression Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 22 / 26

44 MPPP Applications & Model Extensions MPPP 3-Stage Model Conclusions (MPPP) appropriate mathematical tool to describe gene expression averages independent of the chosen distribution Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 22 / 26

45 MPPP Applications & Model Extensions MPPP 3-Stage Model Conclusions (MPPP) appropriate mathematical tool to describe gene expression averages independent of the chosen distribution analytic form formula for any distribution explicit formula depending on the model parameters for specific and interesting distributions Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 22 / 26

46 MPPP Applications & Model Extensions MPPP 3-Stage Model Conclusions Biological consequences: the computed variance may be underestimated deterministic protein elongation might be an upper-bound for protein variance possibility to compute protein variance under more realistic assumptions Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 23 / 26

47 MPPP Applications & Model Extensions MPPP 3-Stage Model Few more results More realistic description of gene expression 4-Stage model and general distribution for protein elongation analysis and proof of the correct assumption for protein degradation (proteolysis/volume dilution) counter-intuitive: var DET (P) > var EXP (P) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 24 / 26

48 MPPP Applications & Model Extensions MPPP 3-Stage Model V. Fromion, E. Leoncini, and P. Robert. Stochastic Gene Expression in Cells: A Point Process Approach. In: SIAM Journal on Applied Mathematics 73.1 (2013), pp Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 25 / 26

49 MPPP Applications & Model Extensions MPPP 3-Stage Model Thanks. Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 26 / 26

50 4-Stage Model One more thing: realistic model of gene expression Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 27 / 26

51 4-Stage Model MPPP 4-Stage model 4-Stage Model: mrna M(t) protein P(t) λ3 F2 (dv ) Emanuele LEONCINI (INRIA) F3 (dy ) Stochastic gene expression Cell Biology - Lyon 28 / 26

52 4-Stage Model MPPP 4-Stage model 4-Stage Model: mrna M(t) ribosome R(t) λ3 F3 (dy ) F4 (dy ) F2 (dv ) Emanuele LEONCINI (INRIA) protein P(t) Stochastic gene expression Cell Biology - Lyon 28 / 26

53 4-Stage Model MPPP 4-Stage model 4-Stage Model: mrna M(t) ribosome R(t) λ3 F3 (dy ) F4 (dy ) F2 (dv ) Emanuele LEONCINI (INRIA) protein P(t) Stochastic gene expression Cell Biology - Lyon 28 / 26

54 4-Stage Model MPPP 4-Stage model 4-Stage Model: mrna M(t) ribosome R(t) λ3 protein P(t) F3 (dy )? µ2 Emanuele LEONCINI (INRIA) µ4 Stochastic gene expression Cell Biology - Lyon 28 / 26

55 4-Stage Model MPPP 4-Stage model 4-Stage Model: 3-Stage model all exponential steps [ var (3) (P) = E [P] 1 + λ ] 3 µ 2 + µ 3 (active gene) 4-Stage model all exponential steps [ ] var (4) λ 3 µ 3 (µ 2 + µ 3 + µ 4 ) (P) = E [P] 1 + (µ 2 + µ 3 )(µ 2 + µ 4 )(µ 3 + µ 4 ) (active gene) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 29 / 26

56 4-Stage Model MPPP 4-Stage model 4-Stage Model: protein elongation lots of identical steps ( 400 amino acids per protein) each step is exponentially distributed The resulting distribution can be described by normal distribution Gamma distribution Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 30 / 26

57 4-Stage Model MPPP 4-Stage model 4-Stage Model: protein elongation lots of identical steps ( 400 amino acids per protein) each step is exponentially distributed The resulting distribution can be described by normal distribution Gamma distribution Problem: hard to obtain explicit formula depending on the model parameters Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 30 / 26

58 4-Stage Model MPPP 4-Stage model 4-Stage Model: protein elongation The resulting distribution can be described by normal distribution Gamma distribution Problem: hard to obtain explicit formula depending on the model parameters Approximation: deterministic protein elongation τ 3 Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 31 / 26

59 4-Stage Model MPPP 4-Stage model 4-Stage Model: 4-Stage model with deterministic elongation [ var D (P) = E(P) 1 + λ 3 + λ ] 2λ 3 (1 δ + )(Λ + µ 2 + µ 4 ). µ 2 + µ 4 (µ 2 + µ 4 )(Λ + µ 2 )(Λ + µ 4 ) Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 32 / 26

60 4-Stage Model MPPP 4-Stage model 4-Stage Model: 4-Stage model with deterministic elongation [ var D (P) = E(P) 1 + λ 3 + λ ] 2λ 3 (1 δ + )(Λ + µ 2 + µ 4 ). µ 2 + µ 4 (µ 2 + µ 4 )(Λ + µ 2 )(Λ + µ 4 ) 4-Stage model with exponential elongation [ var E (P) = E(P) 1 + λ 2 λ 3 (1 δ + )µ 3 µ 2 4 (Λ + µ 2 )(µ 2 4 µ2 3 ) λ 3 µ 3 (µ 2 + µ 3 + µ 4 ) (µ 2 + µ 3 )(µ 2 + µ 4 )(µ 3 + µ 4 ) + ( Λ+µ 2 +µ 3 µ 3 (µ 2 +µ 3 )(Λ+µ 3 ) Λ+µ 2 +µ 4 µ 4 (µ 2 +µ 4 )(Λ+µ 4 ) )]. Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 32 / 26

61 4-Stage Model MPPP 4-Stage model Deterministic vs Exponential Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 33 / 26

62 4-Stage Model MPPP 4-Stage model Conclusions Few results: explicit formula depending on the model parameters for specific assumptions; counter-intuitive: var DET (P) > var EXP (P). Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 34 / 26

63 4-Stage Model MPPP 4-Stage model Conclusions Few results: explicit formula depending on the model parameters for specific assumptions; counter-intuitive: var DET (P) > var EXP (P). Biological consequences: the estimated variance could have been underestimated; deterministic protein elongation as upper-bound for protein variance; possibility to compute (numerically) more precise protein variance with realistic assumptions. Emanuele LEONCINI (INRIA) Stochastic gene expression Cell Biology - Lyon 34 / 26