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Bursting Nucleation
Modelisation probabiliste en biologie cellulaire etmoleculaire
These sous la direction de
M. Adimy, M.C. Mackey & L. Pujo-Menjouet
Romain Yvinec
Institut Camille Jordan - Universite Claude Bernard Lyon 1
Vendredi 05 octobre 2012
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Outline
Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model
Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time
Outline
Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model
Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time
Bursting Nucleation Experiments Reduction Limiting model
Central Dogma
◮ Expression of a gene through transcription/translationprocesses.
◮ Non-linear Feedback regulation.
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Bursting Nucleation Experiments Reduction Limiting model
◮ Bifurcation analysis in Ordinary Differential Equation.
◮ Application to synthetic biology.
[Goodwin, 1965],[Hasty et al., 2001].
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Bursting Nucleation Experiments Reduction Limiting model
Stochasticity in molecular biology
[Eldar and Elowitz, 2010].
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Bursting Nucleation Experiments Reduction Limiting model
Much more accurate measurements
◮ Bifurcation can be studied on probability distributions.
[Song et al., 2010].
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Bursting Nucleation Experiments Reduction Limiting model
Much more accurate measurements
◮ Trajectories can be analyzed on single cells.
[Yu et al., 2006].
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Bursting Nucleation Experiments Reduction Limiting model
New Central dogma
◮ Take into account gene state switching. Interpretation asstochastic processes.
[Berg, 1978],[Peccoud and Ycart, 1995],[Kepler and Elston, 2001],[Paulsson, 2005],[Lipniacki et al., 2006],[Paszek, 2007],
[Shahrezaei and Swain, 2008].
The bursting phenomena
Question 1) When does the stochastic model predict burstphenomenon ?Question 2) What can we say in such cases ?
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Outline
Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model
Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time
Bursting Nucleation Experiments Reduction Limiting model
We consider the following 2d stochastic kinetic chemical reactionmodel (X=’mRNA’, Y=’Protein’)
∅λ1(X ,Y )−−−−−→ X , Production of X at rate λ1(X ,Y )
XNγ1(X ,Y )−−−−−−→ ∅, Destruction of X at rate Nγ1(X ,Y )
∅Nλ2(X ,Y )−−−−−−→ Y , Production of Y at rate Nλ2(X ,Y )
Yγ2(X ,Y )−−−−−→ ∅, Destruction of Y at rate γ2(X ,Y )
with γ1(0,Y ) = γ2(X , 0) = 0 to ensure non-negativity.
BN f (x , y) =λ1(x , y)[
f (x + 1, y)− f (x , y)]
+ Nγ1(x , y)[
f (x − 1, y) − f (x , y)]
+ Nλ2(x , y)[
f (x , y + 1)− f (x , y)]
+ γ2(x , y)[
f (x , y − 1)− f (x , y)]
.
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Bursting Nucleation Experiments Reduction Limiting model
Theorem (R.Y.)If
◮ The degradation function on X satisfies
infx≥1,y≥0
γ1(x , y) = γ > 0.
◮ The production rate of Y satisfies λ2(0, y) = 0, for all y ≥ 0.
◮ λ1 and λ2 are linearly bounded by x + y , and either λ1 or λ2 isbounded.
Then, for all T > 0, (XN(t),Y N(t))t≥0 converges in L1(0,T ) to(0,Y (t)), whose generator is given by
B∞ϕ(y) = λ1(0, y)( ∫ ∞
0
Pt(γ1(1, · )ϕ(· ))(y)dt − ϕ(y))
+ γ2(0, y)[
ϕ(y − 1)− ϕ(y)]
,
Ptg(y) = E[g(Z (t, y)e−
∫t
0γ1(1,Z (s,y))ds
],
Ag(z) = λ2(1, z)(g(z + 1)− g(z)
).
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Bursting Nucleation Experiments Reduction Limiting model
Sketch of the proof
◮ We first show tightness and convergence of X based on
NγE[∫ t
01{XN(s)≥1}ds
]≤ E
[XN(0)
]+E
[∫ t
0λ1(X
N(s),Y N(s))ds].
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Bursting Nucleation Experiments Reduction Limiting model
Sketch of the proof
◮ We first show tightness and convergence of X based on
NγE[∫ t
01{XN(s)≥1}ds
]≤ E
[XN(0)
]+E
[∫ t
0λ1(X
N(s),Y N(s))ds].
◮ We identify the limiting martingale problem
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Bursting Nucleation Experiments Reduction Limiting model
Sketch of the proof
◮ We first show tightness and convergence of X based on
NγE[∫ t
01{XN(s)≥1}ds
]≤ E
[XN(0)
]+E
[∫ t
0λ1(X
N(s),Y N(s))ds].
◮ We identify the limiting martingale problem
λ2(x , y)[
f (x , y+1)−f (x , y)]
+γ1(x , y)[
f (x−1, y)−f (x , y)]
= 0.9/40
Bursting Nucleation Experiments Reduction Limiting model
Sketch of the proof
◮ We first show tightness and convergence of X based on
NγE[∫ t
01{XN(s)≥1}ds
]≤ E
[XN(0)
]+E
[∫ t
0λ1(X
N(s),Y N(s))ds].
◮ We identify the limiting martingale problem
Axg(y) = λ2(x , y)[
g(y + 1)− g(y)]
,
for any x ≥ 1. and we introduce the semigroup Pxt
Pxt g(y) = E
[g(Z x ,y
t )e−∫ t0 γ1(x ,Z
x,ys )ds
].
Now for any bounded function g , define f (0, y) = g(y) and
f (x , y) =
∫ ∞
0Pxt (γ1(x , .)f (x − 1, .))(y)dt.
Then
λ2(x , y)[
f (x , y+1)−f (x , y)]
+γ1(x , y)[
f (x−1, y)−f (x , y)]
= 0.9/40
Bursting Nucleation Experiments Reduction Limiting model
◮ A similar proof for a (continuous state) PDMP model, ofgenerator
Bf (x , y) =− Nγ1(x , y)∂f
∂x+ (Nλ2(x , y)− γ2(x , y))
∂f
∂y
+ λ1(x , y)
∫ ∞
0(f (x + z , y)− f (x , y))h(z)dz .
◮ These proofs are based on a simple idea([Debussche et al., 2011],[Kang and Kurtz, 2011]).
◮ Other proof : reduction on the Fokker-Planck equation.
◮ Different scalings lead to different models.
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Outline
Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model
Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time
Bursting Nucleation Experiments Reduction Limiting model
We look at the stochastic process
dx = −γ(x)dt + dN(λ(x), h(x , ·)),
whose generator is
Af = −γ(x)f ′(x) + λ(x)( ∫ ∞
0f (x + y)h(x , y)dy − f (x)
)
,
and evolution equation on densities
∂u(t, x)
∂t=
∂γ(x)u(t, x)
∂x−λ(x)u(t, x)+
∫ x
0u(t, y)λ(y)h(y , x−y)dy ,
and with
∫ ∞
0h(x , y)dy = 1, for all x .
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Bursting Nucleation Experiments Reduction Limiting model
Probabilistic techniques
If jumps are independent of positions, i.e. h(x , y) = h(y), we have :
Proposition
Suppose x 7→ λ(x) is continuous on (0,∞), λ(0) > 0, γ(x) = γx,E[h]< ∞, and
limx→∞
λ(x)E[h]
γx< 1,
then there exist β < 1, B < ∞ and π (invariant measure) such that
‖P(t, x , ·) − π‖V ≤ BV (x)βt , x ∈ E , t > 0,
where ‖µ‖f = sup|g |≤f | µ(g) | and V (x) = x + 1.
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Bursting Nucleation Experiments Reduction Limiting model
Probabilistic techniques
If jumps are independent of positions, i.e. h(x , y) = h(y), we have :
Proposition
Suppose x 7→ λ(x) is continuous on (0,∞), λ(0) > 0, γ(x) = γx,E[h]< ∞, and
limx→∞
λ(x)E[h]
γx< 1,
then there exist β < 1, B < ∞ and π (invariant measure) such that
‖P(t, x , ·) − π‖V ≤ BV (x)βt , x ∈ E , t > 0,
where ‖µ‖f = sup|g |≤f | µ(g) | and V (x) = x + 1.
◮ Ax = −γx + λ(x)( ∫∞
0 (x + y)h(y)dy − x)
= −(1−λ(x)E
[h
]
γx)γx
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Bursting Nucleation Experiments Reduction Limiting model
Semigroup techniques
∂u(t, x)
∂t︸ ︷︷ ︸
dudt
=∂γ(x)u(t, x)
∂x− λ(x)u(t, x)
︸ ︷︷ ︸
Au=(A0−λ)u
+
∫ x
0u(t, y)λ(y)h(y , x − y)dy
︸ ︷︷ ︸
Bu=J(λu)
(A,D(A)) ⇒ S(t)u(x) = P0(t)u(x)e−
∫t
0λ(φr x)dr
Let C = A+ B. Denote the resolvent RSs u =
∫∞0 e−stS(t)udt.
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Bursting Nucleation Experiments Reduction Limiting model
Semigroup techniques
∂u(t, x)
∂t︸ ︷︷ ︸
dudt
=∂γ(x)u(t, x)
∂x− λ(x)u(t, x)
︸ ︷︷ ︸
Au=(A0−λ)u
+
∫ x
0u(t, y)λ(y)h(y , x − y)dy
︸ ︷︷ ︸
Bu=J(λu)
(A,D(A)) ⇒ S(t)u(x) = P0(t)u(x)e−
∫t
0λ(φr x)dr
Let C = A+ B. Denote the resolvent RSs u =
∫∞0 e−stS(t)udt.
Theorem ([Tyran-Kaminska, 2009])There is a minimal substochastic semigroup P generated by anextension of (C ,D(A)), and which resolvent is given by
RPs u = lim
n→∞
RSs
n∑
k=0
(J(λRSs ))
ku,
and if K = limσ→0 J(λRSσ ) has a unique invariant density, then so
does for P (and P is stochastic).
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Bursting Nucleation Experiments Reduction Limiting model
◮ Under good conditions, K is the transition operator for thediscrete Markov chain“post-jump”, and has for kernel
k(x , y) =
∫ x
0
1{(0,y)}(z)h(z , y − z)λ(z)
γ(z)eQ(x)−Q(z)dz ,
Q(x) =
∫ x
x
λ(z)
γ(z)dz .
◮ Modulo integrability conditions, invariant density v∗ for K andinvariant density u∗ for P are related through
γ(x)u∗(x) =
∫ x
0
H(z , x − z)λ(z)u∗(z)dz , H(z , x) =
∫ ∞
x
h(z , y)dy ,
v∗(x) =
∫ x
0
h(z , x − z)λ(z)u∗(z)dz ,
u∗(x) =1
γ(x)
∫ ∞
x
eQ(y)−Q(x)v∗(y)dy ,
v∗(x) =
∫ x
0
h(z , x − z)λ(z)
γ(z)e−Q(z)
∫ ∞
z
v∗(y)eQ(y)dydz .
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Bursting Nucleation Experiments Reduction Limiting model
Condition for ergodicity in the exponential case
If jumps are independent of positions, i.e. h(x , y) = h(y) andexponentially distributed, of mean b, i.e. h(y) = 1
be−y/b, then
Theorem (M. Tyran-Kaminska, M. Mackey, R.Y.)
Under technical assumptions (for integrability), and if
limx→∞
λ(x)
γ(x)<
1
b,
Q(0) :=
∫ x
0
λ(z)
γ(z)dz = ∞,
then P is ergodic with unique invariant density
u∗(x) =1
cγ(x)e−x/b−Q(x).
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Bursting Nucleation Experiments Reduction Limiting model
Bifurcation
This analytical approach allows us to deduce that the number ofmodes of the stationary state is linked to the solution of
λ(x) =γ(x)
b+ γ′(x).
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Bursting Nucleation Experiments Reduction Limiting model
Further results (not developed here)
◮ This can be used to find λ(x) and b from observations of(u∗, γ).
◮ The convergence rate can be estimated from couplingtechniques.
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Bursting Nucleation Experiments Reduction Limiting model
Further results (not developed here)
◮ This can be used to find λ(x) and b from observations of(u∗, γ).
◮ The convergence rate can be estimated from couplingtechniques.
Perspectives
◮ Other jump size kernel h.
◮ Waiting time properties.
◮ Switch and bursting model.
◮ Include cell division and study population dynamics.
◮ Characterize oscillations in two-dimensional model.
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Outline
Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model
Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time
Bursting Nucleation Prion Model Experiments Nucleation time
Prion Diseases
◮ Creutzfeldt-Jakob : Firsthuman prion diseasedescribed (1929).
◮ Mad cow disease, Scrapie.
◮ Kuru (New Guinea)
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Bursting Nucleation Prion Model Experiments Nucleation time
Prion Diseases
Epidemiology
◮ All prion diseases aretransmissible, i.e. infectious.
◮ Some prion diseases aresporadic, they appearspontaneously, withoutcause.
◮ Some prion diseases aregenetic.
Symptoms
◮ Affect the structure of thebrain ;
◮ Convulsion, Dementia, Lossof balance ;
◮ Always fatal.
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Outline
Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model
Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time
Bursting Nucleation Prion Model Experiments Nucleation time
What is prion ?
A protein
◮ A protein called PRION isthe cause of this disease
◮ It is neither a bacteria, noran viroid like agent !
◮ Stanley Prusiner wasawarded Nobel price inPhysiology and Medicine in1997 for his discovery.
Histopathology
◮ Accumulation of a protein inthe amyloid form.
◮ Spongiosis.
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Bursting Nucleation Prion Model Experiments Nucleation time
Nucleation-Polymerization
Conformation change
Normal ⇔ Misfolded
Lansbury’s model
[Lansbury and Caughey, 1995].
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Outline
Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model
Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time
Bursting Nucleation Prion Model Experiments Nucleation time
◮ In vitro spontaneous polymerization experiments.
◮ Time series of polymer mass.
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Bursting Nucleation Prion Model Experiments Nucleation time
Statistics of nucleation time
◮ Relation between the nucleation time and the initialconcentration in log plots.
◮ Full distribution of the nucleation time.
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Bursting Nucleation Prion Model Experiments Nucleation time
Questions
◮ Can a probabilistic model reproduce the observed variability ?
◮ Can it help to identify parameters ?
◮ Can a model include different strain structures ?
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Outline
Bursting phenomenon in gene expression modelsMolecular biologyTranscriptional/Translational BurstingLimiting model
Nucleation in Prion Polymerization ExperimentsPrion diseasesPrusiner-Lansbury modelIn vitro experimentsStudy of the nucleation time
Bursting Nucleation Prion Model Experiments Nucleation time
Reversible aggregation model
Ci + C1
p−⇀↽−qCi+1
where
Ci = ♯{molecules of size i}.
The nucleation time is given by a waiting time problem,
Tlag = inf{t ≥ 0 : CN(t) = 1},
with initial condition C1(0) = M, Ci(0) = 0, i ≥ 2.N is the nucleus size.
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Bursting Nucleation Prion Model Experiments Nucleation time
Constant monomer formulation
If we supposeC1(t) ≡ M
We can solve exactly the probability distributions (Poisson) and wededuce
S(t) := P{CN(s) = 0, s ≤ t
}= P
{CN(t) = 0
}= e−cN (t),
where (ci )i=2..N are solution of the linear deterministic system (cnis absorbing) :
c2 = pM(12M − c2)− q(c2 − c3),ci = pM(ci−1 − ci )− q(ci − ci+1), 3 ≤ i ≤ N − 2,cN−1 = pM(cN−2 − cN−1)− qcN−1,
cN = pMcN−1.
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Bursting Nucleation Prion Model Experiments Nucleation time
Nucleation time distribution (C1(t) ≡ M)
M → ∞ : Weibull MN
2(N−2)!tN−2 exp(− MN
2(N−1)! tN−1)
q → ∞ : exponential MN
2qN−2 exp(−MN
2qN−2 t)
N = 6,M = 1000,
q =
102,
103,
4.103,
104.
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Bursting Nucleation Prion Model Experiments Nucleation time
Mean nucleation time versus initial monomer quantity inlog scale (C1(t) ≡ M)
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Bursting Nucleation Prion Model Experiments Nucleation time
Nucleation time distribution in log scale (C1(t) ≡ M)
M → ∞ : Weibull MN
2(N−2)!tN−2 exp(− MN
2(N−1)! tN−1)
q → ∞ : exponential MN
2qN−2 exp(−MN
2qN−2 t)
N = 6,M = 1000,
q =
102,
103,
4.103,
104.
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Bursting Nucleation Prion Model Experiments Nucleation time
Mass conservative formulation
We now supposeN∑
i=1
iCi(t) ≡ M.
Kolmogorov backward equations ⇒ Linear system on
S(t, {C 0}) = P{CN(t) = 0 | Ci(0) = C 0
i
}
Problem : Dimension of the system
♯{configuration {C 0},N∑
i=1
iC 0i = M,C 0
N = 0} ≈MN
N!
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Bursting Nucleation Prion Model Experiments Nucleation time
In general, we look for approximate solution for extreme parametervalues : q ≫ M and q ≪ M. We use
◮ known deterministic solution ;
◮ time scale separation ;
◮ scaling laws ;
◮ phase space dimension reduction ;
◮ linear model ;
◮ numerical simulation.
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Bursting Nucleation Prion Model Experiments Nucleation time
Trajectories in the unfavorable case q ≫ M
Pre-equilibrium hypothesis (ex : M = 200,N = 8,q = 1000).
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Bursting Nucleation Prion Model Experiments Nucleation time
Nucleation time distribution in the unfavorable caseq ≫ M : exponential law
Tlag ∼ exponential law, of parameter
< C1CN−1 >t→∞ (M) ≈ c1(t → ∞)cN−1(t → ∞) ≈MN
2qN−2
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Bursting Nucleation Prion Model Experiments Nucleation time
Trajectories in the favorable case M >> q and N large
“Metastable“ trajectory (ex :M = 30000,N = 10,q = 1).
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Bursting Nucleation Prion Model Experiments Nucleation time
Trajectories in the favorable case M >> q and N large
“Metastable“ trajectory. Known phenomenon for the deterministicmodel ([Penrose, 1989],[Wattis, 2006])
1. irreversibleaggregation(up to c∗i )
2. slow ”diffusion”with constantmonomerC1(t) ≡ c∗1
3. convergence toequilibrium
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Bursting Nucleation Prion Model Experiments Nucleation time
Nucleation time distribution in the favorable case M >> q
and N large, c∗N < 1 : bimodal distribution
c∗N < 1 : Linear model with C1 ≡ c∗1 , Ci (0) = c∗i (solid line).
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Bursting Nucleation Prion Model Experiments Nucleation time
Nucleation time distribution in the favorable case M >> q
and N small, c∗N > 1 : Weibull law
c∗N > 1 : Linear model with C1 ≡ M, Weibull law (dashed line).
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Bursting Nucleation Prion Model Experiments Nucleation time
Mean nucleation time versus initial monomer quantity inlog scale : 2 or 3 phases according nucleus size N
1. exponential
2. Linearmodel(startingfrom c∗i )
3. Weibull
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Bursting Nucleation Prion Model Experiments Nucleation time
Mean nucleation time versus initial monomer quantity inlog scale : 2 or 3 phases according nucleus size N
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Bursting Nucleation Prion Model Experiments Nucleation time
Conclusion/Perspectives
◮ Different behavior of the nucleation time
◮ Parameter Identifiability depending on parameter region
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Bursting Nucleation Prion Model Experiments Nucleation time
Conclusion/Perspectives
◮ Different behavior of the nucleation time
◮ Parameter Identifiability depending on parameter region
Perspectives
◮ Different nucleation regime ⇒ Different polymerization regime
◮ Possibility to take into account different polymer structures
◮ Study the nucleation time for size-dependent rates
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Bursting Nucleation Prion Model Experiments Nucleation time
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Bursting Nucleation Prion Model Experiments Nucleation time
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Bursting Nucleation Prion Model Experiments Nucleation time
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