Stochastic single 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.
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Stochastic single gene expression model · Goodwin’s modelStochastic model Reduced modelInverse problemDivision model Can we perform a systematic bifurcation theory on such systems?
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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
Stochastic gene expression modelReduced modelInverse problemFuture work : Taking into account division
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
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.
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
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
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
Inferred bursting rate
x
0 5 10 15
α(x
)#
0
1
2
3
4
5
6
7
8With the exact b
True solution
With b estimated in the same time
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
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
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
Single cell data onself-regulating gene
§ Synthetic Tet-Off in buddingyeast.
§ Feedback modulated by anexternal parameter(doxycycline)
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
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
2
3
4
5
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
Inferred bursting rate
x
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
α(x)
0
10
20
30
40
50
60
70
80
90
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
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
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
Does that Fit ? Can do better !
x
0 1 2 3 4 5 6 7 8 9 10
PDF
0
0.5
1
1.5
2
2.5
3
3.5
4
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
Goodwin’s deterministic model
Stochastic gene expression modelReduced modelInverse problemFuture work : Taking into account division
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
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
10
20
30
40
50
60
0 5 10 15 20 25
X(t)
time
One stochastic trajectory, λb=10
0
10
20
30
40
50
60
0 5 10 15 20 25
X(t)
time
One stochastic trajectory, λb=10
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
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)
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
This may be used to predict the long time behavior of a dividingcell population
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
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...
Goodwin’s model Stochastic model Reduced model Inverse problem Division model
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