Bursting in gene expression model

HAL Id: hal-02797859

https://hal.inrae.fr/hal-02797859

Submitted on 5 Jun 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Bursting in gene expression model

Romain Yvinec, Michael C. Mackey, Marta Tyran-Kamińska, Changjing Zhuge

To cite this version:

Romain Yvinec, Michael C. Mackey, Marta Tyran-Kamińska, Changjing Zhuge. Bursting in gene

expression model. Colloque PDMP, Agence Nationale de la Recherche (ANR). FRA., May 2015,

Saint Martin de Londres, France. �hal-02797859�

Bursting in gene expression model

Romain Yvinec ¹ , Michael C. Mackey ² , Marta Tyran-Kami´ nska ³ , Changjiing Zhuge ⁴ , Jinzhi Lei ⁴

1

BIOS group, INRA Tours, France.

2

McGill University, Canada.

3

University of Silesia, Poland.

4

Tsinghua University, China.

Goodwin’s deterministic model

Stochastic gene expression model

Analytical results on a reduced model

Ongoing work 1) Taking into account division

Ongoing work 2) Recovering the burst statistics

Crick (1958) : Central Dogma.

Jacob, Perrin, S´ anchez, Monod (1960) : Operon.

$

’ ’

’ ’

’ ’

&

’ ’

’ ’

’ ’

% dM

dt “ λ

pE q ´ γ

M , dI

dt “ λ

M ´ γ

I , dE

dt “ λ

I ´ γ

E .

(1)

Goodwin (1965), Griffith (1968), Othmer (1976), Selgrade (1979)...

Goodwin’s model Stochastic model

The transcription rate function λ ₁

§ Inducible Operon : Repressors R interacts with both the Operator O and the Effector E ,

R ` nE é ^K

RE n , K 1 “ RE n

R ¨ E ⁿ , O ` R é ^K

OR , K ₂ “ OR

O ¨ R . With QSSA, and if O tot ! R tot ,

λ ₁ pE q „ O O tot

“ 1 ` K ₁ E ⁿ

1 ` K 2 R tot ` K 1 E ⁿ . (2)

§ Repressible Operon : Similar but

O ` RE ^{n K} é

ORE n , K 2 “ ORE n

O ¨ RE ⁿ . and we get

λ 1 pE q „ O O tot

“ 1 ` K 1 E ⁿ

1 ` pK 1 ` K 2 R tot qE ⁿ . (3)

Bifurcation analysis in ODE

0 5 10 15 20

0 2 4 6 8 10

K κd

uninduced bistable induced

§ Inducible : Mono-stability or Bi-stability.

§ Repressible : Mono-stability or limit cycle.

$

’ ’

’ ’

&

’ ’

’ ’

% dx

dt “ γ

rλ

px

q ´ x

s , dx

dt “ γ

px

´ x

q , dx

dt “ γ

px

´ x

q . (4) Here λ

pxq “ κ

1 ` x

K ` x

.

0 1 2 3 4 5 6 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

x

Goodwin’s model Stochastic model Reduced model With division Inverse problem

Eldar and Elowitz (Nature 2010)

’New’ Central dogma

DNA

mRNA Protein

ON

OFF

Novick & Weiner (1957),Ko et al. (1990),Ozbudak et al. (2002),Elowitz et al. (2002),Raser & O’Shea (2004)...

$

’ ’

’ ’

’ &

’ ’

’ ’

’ %

dx

dt “ G ptqλ ₁ pyptqq ´ γ ₁ xptq , dy

dt “ λ 2 xptq ´ γ 2 yptq , pG “ 0q ÝÝÝÝá âÝÝÝÝ ^αpyptqq

βpyptqq

pG “ 1q .

(5)

Rigney & Schieve (1977),Berg (1978),Peccoud & Ycart (1995),Thattai & Van Oudenaarden (2001)...

Outline

Goodwin’s deterministic model

Stochastic gene expression model

Analytical results on a reduced model

Ongoing work 1) Taking into account division

Ongoing work 2) Recovering the burst statistics

Can we perform a systematic bifurcation theory on such systems ?

§ We are interested in long time behavior.

§ We want to know how many modes has the stationary distribution.

§ This requires in practice ’analytical’ solution.

Goodwin’s model Stochastic model Reduced model With division Inverse problem

A subclass of the ’three-stage’ (DNA, mRNA, Protein) model is the 1D-bursting model (Storage model)

Lf pxq “ ´γpxqf ¹ pxq ` λpxq ż ₈

0

pf px ` yq ´ f pxqqhpx, yqdy (6) where h is the burst size distribution, ş ₈

0 hpx, yqdy “ 1. If hpx, yq “ ´ ^ν

_νpxq ^{px`y q} , ν Œ and ν Ñ 8 0,

Lf pxq “ ´γ pxqf ¹ px q ` λpxq νpxq

ż ₈

x

f ¹ pz qν pzqdz (7) Any stationary distribution satisfies ş ₈

0 Lf pxqu ^˚ pxqdx “ 0, so that ż ₈

0

”

´ γ pxqu ^˚ pxq ` νpxq ż _x

0

λpy q

νpyq u ^˚ py qdy ı

f ¹ pxqdx “ 0 . (8) Hence

u ^˚ pxq “ νpxq C γpxq exp

´ ż _x λpy q γ py q dy

¯

(9)

Bifurcation analysis in SDE

0 1 2 3 4 5 6 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

x

§ Inducible : Uni-modal or Bi-modal.

§ Repressible : Uni-modal.

(under technical assumptions...) upt, xqdx “ P xptq P rx, x ` dx q ( converges as t Ñ 8 (in L ¹ ) towards u ^˚ ,

du

dx “

„

λpxq ´ γp1 ` x bpxq q

 u

pxq γx . with bpxq “ ´

. and γpxq “ γx.

0 2 4 6 8 10 12 14 15

0 5 10 15 20

K

The presence of bursting can drastically alter regions of bistability

κd−/+

b*κb=κd =K bκb−/+ ; b=0.01 bκ_b−/+ ; b=0.1 bκ_b−/+ ; b=1 bκ_b−/+ ; b=10

n=4

κ_d ≡ b*κ_b

κ_d−/+

Goodwin’s model Stochastic model Reduced model With division Inverse problem

Stationary distribution p λ, γ, b q ñ p u ^˚ q

3 3.5 3.8 4 4.44 5 5.5

0 1 2 3 4 5 6 7 8

6

x κb

2.8 3 3.15 3.3 3.7 4 4.45

0 1 2 3 4 5 6 7 8

5

x κ_b

Here λpxq “ κ _b 1 ` x ⁿ

K ` x ⁿ and bpxq “ b (νpxq “ expp´x{bq).

From the 3-stage to the bursting model

$

’ ’

’ ’

’ &

’ ’

’ ’

’ %

dx

dt “ G ptqλ 1 pyptqq ´ γ 1 xptq , dy

dt “ λ ₂ xptq ´ γ ₂ yptq , pG “ 0q ÝÝÝÝá âÝÝÝÝ ^αpyptqq

βpyptqq

pG “ 1q .

(10)

§ If the mRNA lifetime si short (γ ₁ Ñ 8), we can perform an adiabatic reduction (xptq « Gptq ^λ _γ

1

pyptqq) :

$

’ &

’ %

dy

dt “ Gpt q

1

py ptqq ´ γ

y pt q , pG “ 0q ÝÝÝÝá âÝÝÝÝ

βpyptqq

pG “ 1q .

(11)

Goodwin’s model Stochastic model Reduced model With division Inverse problem

$

’ &

’ %

dy

dt “ G ptq

1

py pt qq ´ γ

ypt q , pG “ 0q ÝÝÝÝá âÝÝÝÝ

βpyptqq

pG “ 1q .

(12)

§ If the Gene active periods are short (β Ñ 8), we obtain the bursting model

dy

dt “ Z ptq ´ γ

y ptq , (13) where Z “ ř

i Z i δ T

is a jump process, of jump rate αpyptqq and jump size cumulative distribution of separated form

P ypT

q ě z | y pT

q “ y (

“ exp

´

´ ż

y

γ

β λ

λ

pw qdw

¯ .

Remark

For a constitutive gene, b :“ ^λ _γ

^λ

1

β is the average number of

proteins produced per Gene activation event.

Remark

For the ’2-stage’ model (Telegraph),

$

’ &

’ %

dx

dt “ G ptqλpxptqq ´ γ xptq , pG “ 0q ÝÝÝÝá âÝÝÝÝ ^αpxptqq

βpxptqq

pG “ 1q . (14)

we have (See Boxma et al. 2005) du ^˚

dx “

” αpxq

γx ´ βpxq λpxq ´ γ x

´ λpxq{x ´ γ ` γxpλ ¹ pxq ´ γq{pλpxq ´ γ xq λpxq

ı

u ^˚ (15)

Small recap’ on the forward problem (1/2)

§ We performed an adiabatic reduction to make the problem analytical tractable.

§ We solved the reduced problem for arbitrary coefficients at equilibrium.

§ We performed an (deterministic-analogous) bifurcation study.

0 2.5 5

0 0.5 1 1.5

X

density

0 2 4 6 8 10 0

0.2 0.4 0.6

Y

density

0 5 10

0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 0

0,1 0,2

Y 0 2 4 6 8 10

0 0,1 0,2

Y

0 25 50

0 0.2 0.4 0.6 0.8 1

X

0 2 4 6 8 10 0

0,1 0,2

Y

0 200 400

0 0.2 0.4 0.6 0.8 1

X

γ₁=0.1 γ₁=1 γ₁=10 γ₁=100

Small recap’ on the forward problem (2/2)

§ Careful ! The two notions of deterministic bistability and

’stochastic bistability (bimodality) are in fact quiet different

0 100 200 300 400 500 600 700 800 900 1000

0 0.5 1 1.5 2

2.5 Deterministic Bistabilty

Time x

κ_d=2.3 γ=1 K=4 n=4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

2 4 6 8 10 12

14 Bistability with Bursting Only

Time x

§ (mean) Switching time : can quantify the ’stability’ of each

state.

Outline

Goodwin’s deterministic model

Stochastic gene expression model

Analytical results on a reduced model

Ongoing work 1) Taking into account division

Ongoing work 2) Recovering the burst statistics

Similar results may be obtained for a ’bursting-division’ model.

Lf pxq “ d pxq ż _x

0

pf pyq ´ f pxqqκpx, yqdy

` λpxq ż ₈

0

pf px ` yq ´ f pxqqhpx, y qdy For instance, with uniform repartition kernel (κpx, yq “ 1{x), constant division rate d and constant exponential burst size (hpx, y q “ expp´y{bq),

d dy u ^˚ “

„

´ λ ¹ py q ` d

λpy q ` d ` λpy q λpy q ` d

´ 1 x ` 1

b

¯

´ xb ² bx ` 1 ´ 1

x



u ^˚ pyq

Goodwin’s model Stochastic model Reduced model With division Inverse problem

This may be used to predict the long time behavior of a dividing cell population

scenario 1

A cell tree

Protein

-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.5

Histogram and analytical solution 1

scenario 2

A cell tree

Protein

-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.5

Histogram and analytical solution 1

Outline

Goodwin’s deterministic model

Stochastic gene expression model

Analytical results on a reduced model

Ongoing work 1) Taking into account division

Ongoing work 2) Recovering the burst statistics

Goodwin’s model Stochastic model Reduced model With division Inverse problem

Inverse Problem : p u ^˚ q ñ p λ, γ, b q

For a constitutive gene, we can infer the burst rate (in protein lifetime unit) ^λ _γ and the mean burst size b from the first two (stationary) moments

bλ γ “ E

“ X ‰ ,

b “ Var pX q E

“ X ‰ .

For an auto-regulated gene, we can inverse the formula for the stationary pdf :

pxu ^˚ pxqq ¹

u ^˚ pxq “ λpxq

γ ´ x

bpxq .

Simulated data

Density reconstruction by Kernel Density Estimation

x

0 2 4 6 8 10 12 14 16 18 20

pdf

0 0.05

0.1 0.15 0.2 0.25

True solution Kernel density estimation Histogram

Goodwin’s model Stochastic model Reduced model With division Inverse problem

Inferred bursting rate

protein

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

bursting rate

0 2 4 6 8

10 Recovering Burst rate

Estimated rate with the true b True solution

Estimated rate with a wrong b

Resulting Probability Density Function

protein

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Density

0 0.1 0.2 0.3

Resulting density

True solution KDE Fit with the true b Fit with the wrong b

Goodwin’s model Stochastic model Reduced model With division Inverse problem

Single cell data on self-regulating gene

§ Synthetic Tet-Off in budding yeast.

§ Feedback modulated by an external parameter

(doxycycline)

1) Kernel Density Estimation

YFP Reporter

1 1.5 2 2.5 3 3.5 4 4.5 5

Goodwin’s model Stochastic model Reduced model With division Inverse problem

2) Finding the ’best’ mean burst size (KL distance)

log

(b)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 lo g

(e rr o r) K L

-6 -4 -2 0 2

KL-Error as a fonction of b

3) Inferred burst rate

protein

1 1.5 2 2.5 3 3.5 4 4.5 5

Burst rate

0 50

100 Estimating burst rates

Goodwin’s model Stochastic model Reduced model With division Inverse problem

3) Inferred mean burst size

Experiment

0 2 4 6 8 10 12

0.045 0.05 0.055 0.06 0.065

Burst size value for the Best fit

4) Resulting Probability Density Function

protein

1 1.5 2 2.5 3 3.5 4 4.5 5

density

0 0.5

1 1.5

2

2.5 Fitting with estimating rates

Small recap’ on the inverse problem

§ With the help of the full solution, we obtained a formula to find the parameter functions from the stationary density.

§ We applied this on simulated and real data.

§ The inverse problem is generally ill-posed (cannot find burst size b and burst rate λ at the same time).

§ Although the resulting pdf does usually ’fit’ the data.

§ Work still on progess...

Merci de votre attention !

§ Molecular distributions in gene regulatory dynamics, M.C Mackey, M. Tyran-Kami´ nska and R.Y., Journal of Theoretical Biology (2011) 274 :84-96

§ Dynamic Behavior of Stochastic Gene Expression Models in the Presence of Bursting, M.C Mackey, M. Tyran-Kami´ nska and R.Y., SIAM Journal on Applied Mathematics (2013) 73 :1830-1852

§ Adiabatic reduction of a model of stochastic gene expression

with jump Markov process, R.Y., C. Zhuge, J. Lei, M.C

Mackey, Journal of Mathematical Biology (2014)

68 :1051-1070