Modèles stochastiques d’expression des gènesyvinec.perso.math.cnrs.fr/Talk/RY_2012_roscoffJPP_stochgene.pdfBiologie Moléculaire Modèle Mathématique Transcriptional Bursting

Biologie MoleculaireModele Mathematique

Transcriptional BurstingTranslational Bursting

Le probleme limite

Modeles stochastiques d’expression des genes

Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F.Ramos Marta Tyran-Kaminska and Changjing Zhuge

June 21, 2012

Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes



Le probleme limite

Outline

Biologie Moleculaire

Modele Mathematique

Transcriptional Bursting

Translational Bursting

Le probleme limite




Le probleme limite

Central dogma

[Jacob et al.(1960)Jacob, Perrin, Snchez, and Monod] Loperon:groupe de genes a expression coordonnee par un operateur.




Le probleme limite

Bifurcation analysis in Ordinary Differential Equation, voir[Goodwin(1965)] Oscillatory behavior in enzymatic control processes.Pour une revue,[Hasty et al.(2001)Hasty, McMillen, Isaacs, and Collins]Computational studies of gene regulatory networks: in numero molecular

biology




Le probleme limite

Mise en evidence de la stochasticite en Biologie

[Eldar and Elowitz(2010)] Functional roles for noise in genetic circuits.




Le probleme limite

New Central dogma

DNA

mRNA ProteinON

OFF

[Berg (1978)],[Peccoud(1995)],[Kepler and Elston(2001)],[Paulsson(2005)],

[Shahrezaei and Swain(2008)],[Paszek(2007)],[Lipniacki et al. (2006)]...




Le probleme limite

Much more precise measurement

Bifurcation can be studied on the whole distribution [Becksei et al.

(2012)]




Le probleme limite

The bursting phenomena

Question 1) Quand est-ce que le modele stochastique predit

l’apparition de Burst?

Question 2) Que peut-on dire dans ces cas la?

[Yu et al (2006)]




Le probleme limite

Quelques ordres de grandeurs

[Golding et al. (2005)], [To and Maheshri (2010)] (bacterie) [Raj et al.(2006)], [Schwanhausser et al. (2011)] ( mammalian cells).

Gene stateActivation rate Inactivation rate

λa λi

(min−1) (min−1)0.01 − 10 0.1 − 100

mRNASynthesis rate Degradation rate Transcriptional efficiency

λ1 γ1λ1λi

(min−1) (min−1)

0.1 − 50 10−3-10−1 0.1 − 100Protein

Synthesis rate Degradation rate Transcriptional efficiency

λ2 γ2λ2γ1

(min−1) (min−1)

1 − 50 10−4-10−2 10 − 1000

Table: Parameters involved in the standard model of molecular biology.Note that we give all parameter values in molecule numbers, as they arerequired for stochastic modelisation. For typical cells like E. Coli, 1molecule per cell corresponds roughly([Thattai and van Oudenaarden(2001)]) to a concentration of 1




Le probleme limite

Representation

En version discrete, comme processus de ”naissance” et ”mort”

X0(t) = X0(0) + Y1

(

∫

t

0λa1{X0(s)=0}ka(X2(s))ds

)

− Y2

(

∫

t

0λi1{X0(s)=1}ki (X2(s))ds

)

X1(t) = X1(0) + Y3

(

∫

t

0λ11{X0(s)=1}k1(X2(s))ds

)

− Y4

(

∫

t

0γ1X1(s)ds

)

X2(t) = X2(0) + Y5

(

∫

t

0λ2X1(s)ds

)

− Y6

(

∫

t

0γ2X2(s)ds

)

Ou en version continu, comme ”PDMP” (processus deterministepar morceaux)

X0(t) = X0(0) + Y1

(

∫

t

0λa1{X0(s)=0}ka(x2(s))ds

)

− Y2

(

∫

t

0λi1{X0(s)=1}ki (x2(s))ds

)

x1(t) = 1{X0(t)=1}λ1k1(x2) − γ1x1

x2 = λ2x1 − γ2x2




Le probleme limite

Theorem (Crudu et al (2011) thm 6.1)

On suppose λi = λ1 = n → ∞, et, ki (x1) ≥ β, k1(x1) ≤ x1 + C,ka(x1) ≤ x1 + C Et ka ou k1 est borne

Xn0 (t) = X

n0 (0) + Y1

(

∫

t

0λa1{Xn

0(s)=0}ka(x

n1 (s))ds

)

− Y2

(

∫

t

0n1{Xn

0(s)=1}ki(x

n1(s))ds

)

xn1 (t) =

∫

t

01{Xn

0(s)=1}nk1(x

n1(s))ds−

∫

t

0γ1x

n1(s)ds

alors, dans L1([0,T ], {0, 1}), X n0 ⇒ 0 et xn1 cv vers le proc. stoch.

de generateur

Aϕ(x1) = −γ1x1∂ϕ

∂x1+ λaka(x1)

∫

∞

0

(

ϕ(φ1(t, x1)) − ϕ(x1))

ki (φ1(t, x1))e−

∫ t0 ki (φ1(s,x1))dsdt

pour ϕ ∈ C 1b (R

+) et φ1(t, x1) la solution de

x(t) = x1 +

∫

t

0k1(x(s))ds




Le probleme limite

Idee de preuve

On montre la tension et la convergence de X0 base sur

nE[

∫ t

0

1{Xn0(s)=1}ki(x

n1(s))ds

]

≤ C + λaE[

∫ t

0

ka(xn1 (s))ds

]

E[

xn1 (t)]

≤ nE[

∫ t

0

1{Xn0(s)=1}k1(x

n1(s))ds

]

On identification le probleme martingale limite en prenant commefonction test f (0, x1) = ϕ(x1) et

f (1, x1) =

∫ ∞

0ϕ(φ1(t, x1))ki (φ1(t, x1))e

−∫ t0 ki (φ1(s,x1))dsdt




Le probleme limite

”Bursting” model

On regarde maintenant

x1(t) = x1(0)−

∫ t

0

γ1x1(s−)ds +

∫ t

0

∫ ∞

0

∫ ∞

0

z1{r≤λ1k1(x2(s−))}N(ds, dz , dr)

x2(t) = x2(0) +

∫ t

0

λ2x1(s−)ds −

∫ t

0

γ2x2(s−)ds

avec N une mesure de Poisson d’intensite dsdrh(z)dz , E[

h]

< ∞et la reduction de ce modele quand γ1 → ∞, avec k1 borne.Dans le cas sans regulation

< x1 > =λ1E

[

h]

γ1

< x2 > =λ2λ1E

[

h]

γ2γ1




Le probleme limite

Reduction adiabatique 2

TheoremSi γ1 = λ2 = n → ∞, k1 borne, et, en loi,xn1 (0) → 0,xn2 (0) → x2(0),

xn1 (t) = x

n1 (0) −

∫

t

0nx

n1(s−)ds +

∫

t

0

∫ ∞

0

∫ ∞

0z1{r≤λ1k1(x

n2(s−))}N(ds, dz, dr)

xn2 (t) = x

n2 (0) + n

∫

t

0xn1(s−)ds −

∫

t

0γ2x

n2(s−)ds

alors xn1 → 0 and xn2 → x2 (dans L1([0,T ],R+)) la solution de

x2(t) = x2(0) −

∫

t

0γ2x2(s−)ds +

∫

t

0

∫ ∞

0

∫ ∞

0z1{r≤λ1k1(x2(s−))}N(ds, dz, dr)

avec N une mesure de Poisson d’intensite dsdrh(z)dz, E[

h]

< ∞.




Le probleme limite

Idee de preuve

On montre la tension et la convergence de x1 base sur

E[

xn1 (t)]

≤ E[

xn1 (0)]

− n

∫ t

0

E[

xn1(s)]

ds+ E[

h]

λ1t

E[

xn2 (t)]

≤ E[

xn2 (0)]

+ n

∫ t

0

E[

xn1(s)]

ds

On identification le probleme martingale limite en prenant commefonction test

f (x1, x2) = ϕ(

x2 +

∫ x1

0

λ2(u)

udu

)




Le probleme limite

Reduction adiabatique 3

TheoremSi γ1 = n and hn(z) = 1

nh( z

n), n → ∞, et, en loi,

xn1 (0) → 0,xn2 (0) → x2(0),

xn1 (t) = x

n1 (0) −

∫

t

0nx

n1(s−)ds +

∫

t

0

∫

∞

0

∫

∞

0nz1{r≤λ1k1(x

n2(s−))}N(ds, dz, dr)

xn2 (t) = x

n2 (0) +

∫

t

0xn1 (s−)ds −

∫

t

0γ2x

n2 (s−)ds

alorsxn1n→ 0 and xn2 → x2 (dans L1([0,T ],R+)) la solution de

x2(t) = x2(0) −

∫

t

0γ2x2(s−)ds +

∫

t

0

∫

∞

0

∫

∞

0z1{r≤λ1k1(x2(s−))}N(ds, dz, dr)

avec N une mesure de Poisson d’intensite dsdrh(z)dz.




Le probleme limite

Idee de preuve

On se ramene au cas precedent avec yn1 =xn1n!!

RemarkOn peut montrer qu’il y a convergence sur D(R+,R+) muni de la

topologie de Jakubowski

Pour xn1 , on peut montrer que ∀ε,T > 0, ∃K > 0,

supn

∫ T

0P{

xn1 (s) > K}

ds ≤ ε




Le probleme limite

RemarkOn peut identifier (plus facilement) la limite en utilisant lesfonctionnelles caracteristiques (cas k1 = 1),

Cξ[f ] = E

[

e∫∞0

if (t)ξtdt]

(1)

Pour la mesure de Poisson,

CN [f ] = exp

[

λ1

∫ ∞

0

∫ ∞

0

(e izf (t) − 1)h(z)dzdt

]

,

et (voir [Caceres and Budini(1997)])

Cx1 [f ] = e i f1(0)x01G

N[f1(t)], Cx2 [f ] = e i f2(0)x

02Gx1 [λ2 f2(t)],

avec

fi (t) =

∫ ∞

t

e−γi (s−t)f (s)ds, (i = 1, 2)




Le probleme limite

On regarde le probl‘eme suivant

dx = −γ(x)dt + dN(λ(x), h)

On definit les operateurs suivant sur L1(R+). ∀t ≥ 0,

P0(t)u(x) = u(φ−tx)γ(φ−tx)γ(x) , ie

∫

B

P0(t)u(x)dx =

∫ ∞

0

1{B}(φtx)u(x)dx

et S(t)u(x) = P0(t)u(x)e−

∫

t

0 λ(φr x)dr , ie

S(t)u(x) = eQ(φ−tx)−Q(x)P0(t)u(x)

Q(x) =

∫ x

x

λ(z)

γ(z)dz

enfin J de noyau h,∫

BJu(x)dx =

∫

∞

0

∫

Bh(x, z)u(z)dzdx




Le probleme limite

L’equation d’evolution sur la densite peut se voir comme leprobleme de Cauchy suivant

du

dt= Cu = A0u − λu + J(λu)

avec A0 le generateur de P0. On note A = A0 − λ.

TheoremSi S(t) est un semi-groupe contractant fortement continu(sous-stochastique), alors il existe un semi-groupe Psous-stochastique minimal genere par une extension de (C ,D(A)),de resolvente caracterise par

RPσu = lim

n→∞RSσ

n∑

k=0

(J(λRSσ))

ku

De plus, P est stochastique ssi, pour un σ > 0

limk→∞

||J(λRSσ))

k|| = 0




Le probleme limite

En particulier, si K = limσ→0 J(λRSσ ) est ergodique en moyenne,

alors P est stochastique.

RemarkPour tout u ∈ D(A),

∫

BP(t)u(x)dx =

∫

∞

0P{

x(t) ∈ B, t < t∞|x(0) = x}

u(x)dx

et si Q(0) = ∞, K est stochastique et est l’operateur detransition associe a la chaine de Markov discrete donne par les”position apres saut”.

Pour conclure l’etude asymptotique, on utilisera

LemmaSi P(t) est stochastique et partiellement integrable, et possede uneunique densite invariante alors P(t) est asymptotiquement stable.




Le probleme limite

Application a notre cas

On note G (x) =∫ x

x1

γ(x) et Q(x) =∫ x

xλ(z)γ(z)dz .

Si γ is a continuous function, λ is a nonnegative measurablefunction with λ/γ being locally integrable on (0,∞) and

γ(x) > 0 for x > 0,

∫ x

0

dx

γ(x)= +∞,

∫ x

0

λ(x)

γ(x)dx = +∞,

(2)for some x > 0, S est sous-stochasique, et on peut calculerexplicitement la resolvante

RSσ v(x) =

∫ ∞

x

1

γ(x)eQσ(y)−Qσ(x)v(y)dy

ou Qσ = σG + Q. Finalement K a pour noyau

k(y , x) =

∫ y

0

h(z , x)λ(z)

γ(z)eQ(y)−Q(z)dz




Le probleme limite

Application a notre cas

Si les sauts sont additifs vers le haut, ie h(z , x) = h(x − z) etexponentiellement distribue de moyenne b, un candidat pour ladensite invariante de K est v∗(x) = e−x/b−Q(x), x > 0, et donc un

candidat pour P(t) est u∗ =RS0 v

∗

||RS0 v

∗||= 1

cγ(x)e−x/b−Q(x). Pour

avoir de bonnes conditions d’integrabilite, on peut supposer que(toujours en plus de G (0) = Q(0) = ∞)

limx→∞

λ(x)

γ(x)<

1

b,

et

lim supx→∞

γ(x) > 0, limx→0

e−Q(x)

γ(x)r< ∞, and

∫ δ

0

γ(x)r−1dx < ∞

for some δ, r > 0.




Le probleme limite

Bifurcation

Il est facile de voir que le nombre de maximum de u est donne parles solutions de

λ(x)

γ(x)=

1

b+

γ′(x)

γ(x)2

3

3.5

3.8

4

4.44

5

5.5

0 1 2 3 4 5 6 7 86

x

κb




Le probleme limite

Travail en cours

I Probleme inverse

I Taux de convergence

I Modele ”switch+bursting”

I Caracteriser les oscillations dans le modele 2D

I Dynamique de population (division cellulaire)




Le probleme limite

merci

merci de votre attention




Le probleme limite

References I

O. G. Berg.A model for the statistical fluctuations of protein numbers in amicrobial population.Journal of Theoretical Biology, 71(4):587–603, 1978.ISSN 0022-5193.doi: DOI:10.1016/0022-5193(78)90326-0.Model de bursting, un ARN messager est traduitinstantantment en r proteine avant de degrader. Burstgeometrique. Taux de production d’ARNm suivant unprocessus de poisson. Possibilit d’avoir diffrentes copies degnes, et des instants de production dterministes. Pas defeedback. Degradation des proteins par dilution. Division descellules, rpartition binomiale. Distribution du nombre de




Le probleme limite

References II

proteine dans le cas de steady-state growth. Distribution nonPoissonienne, calcul des moments. Application a des donneessur le lac repressor et beta-galactose.

M. O. Caceres and A. A. Budini.The generalized ornstein - uhlenbeck process.Journal of Physics A: Mathematical and General, 30(24):8427–, 1997.ISSN 0305-4470.URL http://stacks.iop.org/0305-4470/30/i=24/a=009.


http://stacks.iop.org/0305-4470/30/i=24/a=009



Le probleme limite

References III

A. Crudu, A. Debussche, A. Muller, and O. Radulescu.Convergence of stochastic gene networks to hybrid piecewisedeterministic processes.To appear in Annals of Applied Prob., 2011.

A. Eldar and M. B. Elowitz.Functional roles for noise in genetic circuits.Nature, 467(7312):167–73, Sept. 2010.ISSN 1476-4687.doi: 10.1038/nature09326.URL http://www.ncbi.nlm.nih.gov/pubmed/20829787.Review of the functional role of noise for cell properties. Lotsof discussion of stochastic differentiation.


http://www.ncbi.nlm.nih.gov/pubmed/20829787



Le probleme limite

References IV

I. Golding, J. Paulsson, S. Zawilski, and E. Cox.Real-time kinetics of gene activity in individual bacteria.Cell, 123:1025–1036, 2005.Real time mesurement in E.Coli. Burst size, ON and OFFperiod measured. Discussion of the origin of burst.

B. C. Goodwin.Oscillatory behavior in enzymatic control processes.Advances in Enzyme Regulation, 3:425 – 428, IN1–IN2,429–430, IN3–IN6, 431–437, 1965.ISSN 0065-2571.doi: DOI:10.1016/0065-2571(65)90067-1.




Le probleme limite

References V

Oscillations are proved for a 2d or 3d ODE model of geneexpression with negative feedback. Biological parametersvalues are given from litterature.

J. Hasty, D. McMillen, F. Isaacs, and J. J. Collins.Computational studies of gene regulatory networks: in numeromolecular biology.Nat Rev Genet, 2(4):268–279, 2001.URLhttp://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?c

Review of modelling strategies of gene expression network andexperimentally constructed synthetic network. Discussion onmany examples of Operon. Many references on the historical


http://eutils.ncbi.nlm.nih.gov/entrez/eutils/elink.fcgi?cmd=prlinks&dbfrom=pubmed&retmode=ref&id=11283699



Le probleme limite

References VI

main papers for both modelling strategies and experimentalfindings.

C. Hsu, S. Scherrer, A. Buetti-Dinh, P. Ratna, J. Pizzolato,V. Jaquet, and A. Becskei.Stochastic signalling rewires the interaction map of a multiplefeedback network during yeast evolution.Nature communications, 3:682–682, Jan. 2012.URLhttp://www.pubmedcentral.nih.gov/articlerender.fcgi?artid

Data pour fitter.


http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3293423&tool=pmcentrez&rendertype=abstract



Le probleme limite

References VII

F. Jacob, D. Perrin, C. Snchez, and J. Monod.L’operon: groupe de genes a expression coordonnee par unoperateur.C. R. Acad. Sci. Paris, 250:1727–1729, 1960.The definition of Operon is given as a groupe de gene.repressor and interactions between gene are proved in thelactose system.




Le probleme limite

References VIII

T. Kepler and T. Elston.Stochasticity in transcriptional regulation: Origins,consequences, and mathematical representations.Biophy. J., 81:3116–3136, 2001.ON,OFF and protein model in a discrete or continuoussettings, with and without feedback. Langevin approx.Analytical formula and bifurcation study. Escape time. Mutualrepressor system.




Le probleme limite

References IX

T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A. R. Brasier,and M. Kimmel.Transcriptional stochasticity in gene expression.Journal of Theoretical Biology, 238(2):348–367, 2006.ISSN 0022-5193.URLhttp://www.sciencedirect.com/science/article/B6WMD-4GP1VW

On-Off model with continuous state-space. Linear regulationon a single gene model, and other simple regulatory network.Analytic formulas for asymptotic distribution with a QSSAassumption, and numerical simulations in other cases.


http://www.sciencedirect.com/science/article/B6WMD-4GP1VW1-2/2/e244dd3501617e5d901e99b0c2424e81



Le probleme limite

References X

P. Paszek.Modeling stochasticity in gene regulation: characterization inthe terms of the underlying distribution function.Bulletin of Mathematical Biology, 69(5):1567–601, July 2007.ISSN 0092-8240.doi: 10.1007/s11538-006-9176-7.URL http://www.ncbi.nlm.nih.gov/pubmed/17361363.On-Off single gene, moment calculus in a discrete model,continuous and hybrid approximation. no feedback. Extensivenumerical simulations to distribution function.





Le probleme limite

References XI

J. Paulsson.Models of stochastic gene expression.Physics of Life Reviews, 2(2):157–175, June 2005.ISSN 15710645.doi: 10.1016/j.plrev.2005.03.003.URLhttp://linkinghub.elsevier.com/retrieve/pii/S157106450500

Review of On-Off model without regulation, gene state,mRNA, protein, effective protein and so on. Deterministicformulation, stochastic one. Moment and noise calculation,interpretation in terms of burst. Noise calculation followseither by exact moment equations or Fluctuation-Dissipation


http://linkinghub.elsevier.com/retrieve/pii/S1571064505000138



Le probleme limite

References XII

Theorem (which is the LNA, van Kampen etc...). Discussionon noise, and short review of past stochastic models.

J. Peccoud.Markovian modelling of Gene Product Synthesis.pdf.Theor. Popul. Biol., 48:224–234, 1995.Time-dependent and asymptotic solution of an on-off/proteinmodel without regulation. Hints to prove ergodicity andexponential convergence are given. Estimators of parametersusing either distribution or moment statistics is discuted.




Le probleme limite

References XIII

A. Raj, C. Peskin, D. Tranchina, D. Vargas, and S. Tyagi.Stochastic mRNA synthesis in mammalian cells.PLoS Biol., 4:1707–1719, 2006.Measures of mRNA level in Mamalian cells. Model ON,OFF,mRNA in discrete and continuous settings. No feedback.Discussion of burst occurence and of possible mechanisms bywhich the cell can regulate mRNA transcription (geneactivation, de-activation, transcription rate). It is found thatthe burst size is regulated rather than the burst frequency.




Le probleme limite

References XIV

B. Schwanhausser, D. Busse, N. Li, G. Dittmar,J. Schuchhardt, J. Wolf, W. Chen, and M. Selbach.Global quantification of mammalian gene expression control.Nature, 473(7347):337–42, May 2011.ISSN 1476-4687.doi: 10.1038/nature10098.URL http://www.ncbi.nlm.nih.gov/pubmed/21593866.Absolute count of mRNA and protein level over 5000 genes inMamalian cells, deduction from the model of transcription,translation rate and half-live times. Many parameters values.





Le probleme limite

References XV

V. Shahrezaei and P. Swain.Analytic distributions for stochastic gene expression.Proc. Nat. Acad. Sci, 105:17256–17261, 2008.3 stage model with analytical time dependent and stationarydistribution. Reduction techniques with the characteristicmethod on the Fokker-Planck equation.

M. Thattai and A. van Oudenaarden.Intrinsic noise in gene regulatory networks.Proceedings of the National Academy of Sciences of theUnited States of America, 98(15):8614–8619, 2001.URLhttp://www.pnas.org/content/98/15/8614.abstract.


http://www.pnas.org/content/98/15/8614.abstract



Le probleme limite

References XVI

mRNA/protein model of a single gene with autoregulation.Linearized regulation provide approximation. Time-dependentMoments calculation, and idea of bursting. Some parametervalues compared to data.

T.-L. To and N. Maheshri.Noise can induce bimodality in positive transcriptionalfeedback loops without bistability.Science (New York, N.Y.), 327(5969):1142–5, Feb. 2010.ISSN 1095-9203.doi: 10.1126/science.1178962.URL http://www.ncbi.nlm.nih.gov/pubmed/20185727.Experimental measurement of a positive regulated single gene.There is no cooperativy but bimodality is observed, as result of





Le probleme limite

References XVII

fluctuations of TF, and bursting. In suppl mat, discussion ofthe various model 2 stage, 3 stage, discrete, continuousetc...Dsitribution of on-off model with non-linear regulation.Regulation either on the activation rate (then of the burstfrequency) or the deactivation rate (the non the burst size).Burst size regulation is also considered.

J. Yu, J. Xiao, X. Ren, K. Lao, and X. Xie.Probing gene expression in live cells, one protein molecule at atime.Science, 311:1600–1603, 2006.real time measurement, burst kinetics in E.Coli.


Documents

Modèles stochastiques d’expression des gènesyvinec.perso.math.cnrs.fr/Talk/RY_2012_roscoffJPP_stochgene.pdfBiologie Moléculaire Modèle Mathématique Transcriptional Bursting