Stochastic single gene expression model · Goodwin’s modelStochastic model Reduced modelInverse...

Stochastic single gene expression model

Romain Yvinec1, Michael C. Mackey2, MartaTyran-Kaminska3, Changjiing Zhuge4, Jinzhi Lei4

1BIOS group, INRA Tours, France.

2McGill University, Canada.

3University of Silesia, Poland.

4Tsinghua University, China.

Goodwin’s deterministic model

Stochastic gene expression modelReduced modelInverse problemFuture work : Taking into account division

Goodwin’s model Stochastic model

Crick (1958) : Central Dogma.Jacob, Perrin, Sanchez, Monod (1960) : Operon.

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dM

dt“ λ1pE q ´ γ1M ,

dI

dt“ λ2M ´ γ2I ,

dE

dt“ λ3I ´ γ3E .

(1)

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


The transcription rate function λ1

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

R ` nEK1é REn , K1 “

REn

R ¨ En,

O ` RK2é OR , K2 “

OR

O ¨ R.

With QSSA, and if Otot ! Rtot ,

λ1pE q „O

Otot“

1` K1En

1` K2Rtot ` K1En. (2)

§ Repressible Operon : Similar but

O ` REn K2é OREn , K2 “

OREn

O ¨ REn.

and we get

λ1pE q „O

Otot“

1` K1En

1` pK1 ` K2RtotqEn. (3)


Bifurcation analysis in ODE

0 5 10 15 200

2

4

6

8

10

K

κd

uninduced

bistable

induced

§ Inducible : Mono-stabilityor Bi-stability.

§ Repressible : Mono-stabilityor limit cycle.

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dx1dt

“ γ1rf px3q ´ x1s ,

dx2dt

“ γ2px1 ´ x2q ,

dx3dt

“ γ3px2 ´ x3q .

(4)

Here f pxq “ κd1` xn

K ` xn.

0 1 2 3 4 5 6 70

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0.9

1

1.1

x

Goodwin’s model Stochastic model Reduced model Inverse problem Division model

Eldar and Elowitz (Nature 2010)


’New’ Central dogma

DNA

mRNA ProteinON

OFF

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

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dx

dt“ G ptqλ1pyptqq ´ γ1xptq ,

dy

dt“ λ2xptq ´ γ2yptq ,

pG “ 0qαpyptqqÝÝÝÝáâÝÝÝÝβpyptqq

pG “ 1q .

(5)

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

Outline




Can we perform a systematic bifurcation theory on suchsystems ?

§ If the gene switching rates are fast (α, β Ñ8), we recoverthe Goodwin’s model (no gene noise).

§ If the mRNA lifetime si short (γ1 Ñ8), we can perform an

adiabatic reduction (xptq « G ptqλ1γ1pyptqq) :

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dy

dt“ G ptqλ2λ1

γ1pyptqq ´ γ2yptq ,


pG “ 1q .(6)


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dy

dt“ G ptqλ2λ1

γ1pyptqq ´ γ2yptq ,


pG “ 1q .(7)

§ If the Gene active periods are short (β Ñ8), we obtain thebursting model

dy

dt“ Z ptq ´ γ2yptq , (8)

where Z “ř

i ZiδTiis a jump process, of jump rate αpyptqq

and jump size distribution

P

ypT`i q ě z | ypT´i q “ y(

“ exp´

´

ż z

y

γ1β

λ1λ2pwqdw

¯

.

RemarkFor a constitutive gene, b :“ λ1λ2

γ1βis the average number of

proteins produced per Gene activation event.


Bifurcation analysis inSDE

0 1 2 3 4 5 6 70

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1

1.1

x

§ Inducible :Uni-modal orBi-modal.

§ Repressible :Uni-modal.

(under technical assumptions...)upt, yqdy “ P

yptq P ry , y ` dyq(

converges as t Ñ8 (in L1) towards u˚,

du˚

dy“

„

αpyq ´ γ2p1`y

bpyqq

u˚pyq

γ2y.

Here αpyq “ κb1` yn

K ` yn, and bpyq “ b .

0 2 4 6 8 10 12 14 150

5

10

15

20

K

The presence of bursting can drastically alter regions of bistability

κd−/+

b*κb=κ

d =K

bκb−/+

; b=0.01

bκb−/+

; b=0.1

bκb−/+

; b=1

bκb−/+

; b=10

n=4

κd ≡ b*κ

bκ

d−/+


Stationary distribution pα, γ, bq ñ pu˚q

3

3.5

3.8

4

4.44

5

5.5

0 1 2 3 4 5 6 7 86

x

κb

2.8

3

3.15

3.3

3.7

4

4.45

0 1 2 3 4 5 6 7 85

x

κb

Here αpyq “ κb1` yn

K ` ynand bpyq “ b.

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

§ We performed an adiabatic reduction to make the problemanalytical tractable.

§ We solved the reduced problem for arbitrary coefficients atequilibrium.

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

0 2.5 50

0.5

1

1.5

X

density

0 2 4 6 8 10

0

0.2

0.4

0.6

Y

density

0 5 100

0.2

0.4

0.6

0.8

1

X

0 2 4 6 8 10

0

0,1

0,2

Y0 2 4 6 8 10

0

0,1

0,2

Y

0 25 500

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0.4

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1

X

0 2 4 6 8 10

0

0,1

0,2

Y

0 200 4000

0.2

0.4

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0.8

1

X

γ1=0.1 γ

1=1 γ

1=10 γ

1=100

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

§ Careful ! The two notions of deterministic bistability andstochastic bimodality are in fact quiet different

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5Deterministic Bistabilty

Time

x

κd=2.3

γ=1

K=4n=4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

2

4

6

8

10

12

14Bistability with Bursting Only

Time

x

Outline




Inverse Problem :pu˚q ñ pα, γ, bq

For a constitutive gene, we can infer the burst rate (in proteinlifetime unit) α

γ and the mean burst size b from the first twomoments

bα

γ“ E

“

X‰

,

b “VarpX q

E“

X‰ .

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

pxu˚pxqq1

u˚pxq“αpxq

γ´

x

bpxq.


Simulated data

Density reconstruction by Kernel Density Estimation

x

0 2 4 6 8 10 12 14 16 18 20

pd

f

0

0.05

0.1

0.15

0.2

0.25True solution

Kernel density estimation

Histogram


Inferred bursting rate

x

0 5 10 15

α(x

)#

0

1

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3

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8With the exact b

True solution

With b estimated in the same time


Resulting Probability Density Function

x

0 5 10 15

pdf

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35True solution

With the exact b

With b estimated at the same time

Kernel Density estimation


Single cell data onself-regulating gene

§ Synthetic Tet-Off in buddingyeast.

§ Feedback modulated by anexternal parameter(doxycycline)


Single cell data onself-regulating gene

Smoothed pdf obtained bykernel density estimation

x

1 1.5 2 2.5 3 3.5 4 4.5 5

pdf

0

1

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4

5


Inferred bursting rate

x

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

α(x)

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90


Resulting Probability Density Function

x

0 5 10 15 20 25 30

PDF

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Does that Fit ? Can do better !

x

0 1 2 3 4 5 6 7 8 9 10

PDF

0

0.5

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1.5

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2.5

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3.5

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4.5

5

Small recap’ on the inverse problem

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

§ We applied this on simulated and real data.

§ The inverse problem seems generally ill-posed (cannot findburst size b and burst rate α at the same time).

§ If the resulting pdf does not ’fit’ the data, it may be a goodreason to rule out this model.

§ Work still on progess...

Outline




Similar results may be obtained for a continuous growth divisionmodel or a bursting-division model. For instance, with uniformrepartition kernel, constant division rate d and constant burst sizeb,

d

dyu˚ “

„

´α1pyq ` d

αpyq ` d`

αpyq

αpyq ` d

´1

x`

1

b

¯

´xb2

bx ` 1´

1

x

u˚pyq

0

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30

40

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60

0 5 10 15 20 25

X(t)

time

One stochastic trajectory, λb=10

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30

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60

0 5 10 15 20 25

X(t)

time

One stochastic trajectory, λb=10


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

A cell tree, λb=10 A cell tree, λb=100

§ Inverse problems may be adapted to genealogical tree data(Doumic et al. 2014, Krell 2014)


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

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0 10 20 30 40 50 60

u*(x)

x

Steady-state profile

λb=1, b=10

λb=5, b=2

λb=10, b=1

λb=100, b=0.1

§ Such framework can be used to study non-linear populationmodel...


Vielen Dank !

§ Molecular distributions in gene regulatory dynamics, M.CMackey, M. Tyran-Kaminska and R.Y., Journal of TheoreticalBiology (2011) 274 :84-96

§ Dynamic Behavior of Stochastic Gene Expression Models inthe Presence of Bursting, M.C Mackey, M. Tyran-Kaminskaand R.Y., SIAM Journal on Applied Mathematics (2013)73 :1830-1852

§ Adiabatic reduction of a model of stochastic gene expressionwith jump Markov process, R.Y., C. Zhuge, J. Lei, M.CMackey, Journal of Mathematical Biology (2014)68 :1051-1070

Stochastic single gene expression model · Goodwin’s modelStochastic model Reduced modelInverse...

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