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
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
Le probleme limite
Outline
Biologie Moleculaire
Modele Mathematique
Transcriptional Bursting
Translational Bursting
Le probleme limite
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
Le probleme limite
Central dogma
[Jacob et al.(1960)Jacob, Perrin, Snchez, and Monod] Loperon:groupe de genes a expression coordonnee par un operateur.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
Le probleme limite
Mise en evidence de la stochasticite en Biologie
[Eldar and Elowitz(2010)] Functional roles for noise in genetic circuits.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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)]...
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
Le probleme limite
Much more precise measurement
Bifurcation can be studied on the whole distribution [Becksei et al.
(2012)]
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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)]
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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]
< ∞.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
)
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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)
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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)
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
Le probleme limite
merci
merci de votre attention
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
Biologie MoleculaireModele Mathematique
Transcriptional BurstingTranslational Bursting
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.
Romain Yvinec Jinzhi Lei Michael C. Mackey Alexandre F. Ramos Marta Tyran-Kaminska and Changjing ZhugeModeles stochastiques d’expression des genes
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