Stochastic models of protein production with feedback Renaud - - PowerPoint PPT Presentation

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Stochastic models of protein production with feedback

Renaud Dessalles joint work with Vincent Fromion and Philippe Robert

INRA Jouy-en-Josas - INRIA Rocquencourt (Fance)

• 16th. May 2016

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Presentation

Biological context Mathematical framework Equilibrium results Other aspects of the controlled model

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Part 1 Biological context

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Cells and proteins

▶ Cells: unit of life. ▶ Its goal: grow and divide. ▶ Functional molecules:

proteins

▶ enzymes, wall, energy,

etc.

▶ Produced from the genes

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Protein production: A central mechanism

Proteins represents:

▶ 50% of the dry mass ▶ ∼ 3 million molecules ▶ ∼ 2000 different types ▶ from few dozens up to 105 proteins per type

It needs to be duplicated in one cell cycle (approx. 30 min)

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Protein production: A central mechanism

Proteins represents:

▶ 50% of the dry mass ▶ ∼ 3 million molecules ▶ ∼ 2000 different types ▶ from few dozens up to 105 proteins per type

It needs to be duplicated in one cell cycle (approx. 30 min) 85% of the resources for protein production

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Classic protein production mechanism

Protein production in 3 steps:

• 1. Gene regulation
• 2. Transcription: to produce mRNA
• 3. Translation: to produce proteins

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Highly variable process

The protein production subject to high variability:

▶ Interior of bacteria: non-organized medium ▶ Mobility of compounds: through random diffusion ▶ Cellular mechanism: random collision between molecules

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Highly variable process

The protein production subject to high variability:

▶ Interior of bacteria: non-organized medium ▶ Mobility of compounds: through random diffusion ▶ Cellular mechanism: random collision between molecules

Problem: 85% of the resources for the protein production, impacted by a large variability.

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Highly variable process

The protein production subject to high variability:

▶ Interior of bacteria: non-organized medium ▶ Mobility of compounds: through random diffusion ▶ Cellular mechanism: random collision between molecules

Problem: 85% of the resources for the protein production, impacted by a large variability. A main issue for the bacteria: control the variability in protein production.

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Protein production mechanism with feedback

Production with feedback: the protein binds to its own gene.

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More proteins ⇒ Gene more inactive A way to reduce variability?

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Comparison of models

Classical production vs Feedback production

▶ Conjecture: less variability in proteins with feedback

production.

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Comparison of models

Classical production vs Feedback production

▶ Conjecture: less variability in proteins with feedback

production.

Our Goal

Comparison of distributions of proteins in the two models.

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Part 2 Mathematical framework

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Markovian description

Framework for protein production modeling:

▶ Rigney and Schieve (1977) ▶ Berg (1978) ▶ Paulsson (2005)

Three types of events:

▶ Encounter between molecules ▶ Elongation of molecules ▶ Lifetime of molecules

Assumption: Exponential times

Each event occurs at exponential time.

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The classical model

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I = 1 I = 0 λ+

1

λ−

1

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The classical model

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I = 1 M I = 0 ∅ λ+

1

λ−

1

λ2I µ2M

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The classical model

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I = 1 M P I = 0 ∅ ∅ λ+

1

λ−

1

λ2I µ2M λ3M µ3P

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Mean and variance for the classical model

For the classical model, the mean and the variance are known

Paulsson (2005):

▶ Equality of flows gives

E [P] = λ+

1

λ+

1 + λ− 1

· λ2 µ2 · λ3 µ3

▶ Equilibrium equations give:

Var [P] = E [P] ( 1 + λ3 µ2 + µ3 +λ2λ3 ( 1 − λ+

1 /

( λ+

1 + λ− 1

)) ( λ+

1 + λ− 1 + µ2 + µ3

) (µ2 + µ3) ( λ+

1 + λ− 1 + µ2

) ( λ+

1 + λ− 1 + µ3

) ) .

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The feedback model

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IF = 1 MF PF IF = 0 ∅ ∅ λ+

1

• λ−

1 PF

λ2IF µ2MF λ3MF µ3PF

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Mean and variance for the Feedback model

▶ Equality of flows gives

E [PC] = E [IC] · λ2 µ2 · λ3 µ3 .

▶ Problem : no known expression for E [IC]:

▶ the equality of flows on IC:

• λ−

1 E [ICPC] = λ+ 1 (1 − E [IC]) .

Difficulties to make comparisons between the two models

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Part 3 Equilibrium results

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Scaling

▶ Introduction of a scaling:

Gene regulation timescale Messenger RNA timescale } faster than the protein time scale

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Scaling

▶ Introduction of a scaling:

Gene regulation timescale Messenger RNA timescale } faster than the protein time scale

IN

F = 1

M N

F

P N

F

IN

F = 0

∅ ∅ Nλ+

1

N λ−

1 P N F

Nλ2IN

F

Nµ2M N

F

λ3M N

F

µ3P N

F

N scaling parameter

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Effects of the scaling

Example: Feedback model

▶ State of the model:

( I N

F (t), MN F (t), PN F (t)

)

▶ I N

F and MN F on a quick timescale.

▶ PN

F on a slow timescale.

I N

F and MN F reach some equilibrium depending on the

slow current PN

F (t) state

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Convergence of the gene regulation and the messengers

τ N

1 : the first time of jump of PN F ;

Starting at number of proteins x = PN

F (0) ;

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Convergence of the gene regulation and the messengers

τ N

1 : the first time of jump of PN F ;

Starting at number of proteins x = PN

F (0) ; ▶ (

I N

F (t)

) reaches its equilibrium quickly: E [ I N

F (t)|0 < t < τ N 1 , PN F (0) = x

] N→∞ − − − − → λ+

1

λ+

1 +

λ−

1 x

.

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Convergence of the gene regulation and the messengers

τ N

1 : the first time of jump of PN F ;

Starting at number of proteins x = PN

F (0) ; ▶ (

I N

F (t)

) reaches its equilibrium quickly: E [ I N

F (t)|0 < t < τ N 1 , PN F (0) = x

] N→∞ − − − − → λ+

1

λ+

1 +

λ−

1 x

.

▶ (

MN

F (t)

) reaches its equilibrium quickly: E [ MN

F (t)|0 < t < τ N 1 , PN F (0) = x

] N→∞ − − − − → λ2 µ2 · λ+

1

λ+

1 +

λ−

1 x

.

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Convergence of the gene regulation and the messengers

τ N

1 : the first time of jump of PN F ;

Starting at number of proteins x = PN

F (0) ; ▶ (

I N

F (t)

) reaches its equilibrium quickly: E [ I N

F (t)|0 < t < τ N 1 , PN F (0) = x

] N→∞ − − − − → λ+

1

λ+

1 +

λ−

1 x

.

▶ (

MN

F (t)

) reaches its equilibrium quickly: E [ MN

F (t)|0 < t < τ N 1 , PN F (0) = x

] N→∞ − − − − → λ2 µ2 · λ+

1

λ+

1 +

λ−

1 x

.

▶ Rate of production of proteins tends to:

λ3 · λ2 µ2 · λ+

1

λ+

1 +

λ−

1 x

.

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Convergence of the models

Theorem

The process ( PN

F (t)

) converges in distribution to a birth and death process with (x number of proteins): βx = λ3 · λ2 µ2 · λ+

1

λ+

1 +

λ−

1 x

and δx = µ3x.

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Convergence of the models

Theorem

The process ( PN

F (t)

) converges in distribution to a birth and death process with (x number of proteins): βx = λ3 · λ2 µ2 · λ+

1

λ+

1 +

λ−

1 x

and δx = µ3x. Idem for uncontrolled model:

Theorem

The process ( PN(t) ) converges in distribution to a birth and death process with (x number of proteins): βx = λ3 · λ2 µ2 · λ+

1

λ+

1 + λ− 1

and δx = µ3x.

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Feature of the scaled models

▶ Equilibrium distributions.

▶ Classical model: P∞ follow a Poisson distribution:

P∞ ∼ P (λ3 µ3 · λ2 µ2 · λ+

1

λ+

1 + λ− 1

)

▶ Feedback model: P∞ follow the limit distribution

πF(x) = 1 Z · x! (λ3 µ3 · λ2 µ2 )x x−1 ∏

i=0

λ+

1

λ+

1 +

λ−

1 i

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Feature of the scaled models

▶ Equilibrium distributions.

▶ Classical model: P∞ follow a Poisson distribution:

P∞ ∼ P (λ3 µ3 · λ2 µ2 · λ+

1

λ+

1 + λ− 1

)

▶ Feedback model: P∞ follow the limit distribution

πF(x) = 1 Z · x! (λ3 µ3 · λ2 µ2 )x x−1 ∏

i=0

λ+

1

λ+

1 +

λ−

1 i

Variance comparison

Var [P∞] = E [P∞] and Var [P∞

F ] ≤ E [P∞ F ]

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Asymptotic behaviour of the Feedback model

Introducing: ρ := λ+

1

• λ−

1

λ3 µ3 · λ2 µ2 and η := λ+

1

• λ−

1

− 1 it comes: πF(x) = 1 Z · x!ρx

x

∏

i=1

1 η + i .

Asymptotic behaviour

Increase ρ while keeping η constant: Increase protein production while keeping the gene regulation constant

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Asymptotic behaviour of the Feedback model

With Laplace method:

Theorem

Convergence in distribution: lim

ρ→∞

P∞

F − aρ

√aρ = N ( 0, 1/ √ 2 ) with aρ = (√ η2 + 4ρ − η ) .

Corollary

lim

ρ→∞

E [P∞

F ]

√ρ = 1 and lim

ρ→∞

Var [P∞

F ]

E [ P∞

F

] = 1 2

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Conclusion for the noise control

For the scaled model: Var [P∞] = E [P∞] and Var [P∞

F ] ≤ E [P∞ F ]

Asymptotic behaviour: Var [P∞] = E [P∞] and Var [P∞

F ]

∼

ρ→∞

1 2E [P∞

F ]

The reduction of variance is limited in feeback model.

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Part 4 Other aspects of the controlled model

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Dynamical aspects

Equilibrium reaching

Which model go faster to reach the equilibrium? Biological example: need for a quick activation of the protein production.

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Dynamical aspects

Equilibrium reaching

Which model go faster to reach the equilibrium? Biological example: need for a quick activation of the protein production. Simulations: comparison for the two models:

▶ Starting at a low protein production ▶ Evolution to a high level of protein production

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Simulation for dynamical aspects

▶ 1000 simulations

▶ Thick lines: average of

trajectories

▶ Fine lines: ± standard

deviation

▶ Here, controlled model is

20% faster

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Thank you for you attention

Article:

Dessalles, R., Fromion, V., and Robert, P. (2015). arXiv:1509.02045

PhD work supervised by

▶ Vincent Fromion ▶ Philippe Robert

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Other work

▶ Regulation on the mRNA rather than on the gene

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▶ Intermediate metabolite step in regulation