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Bursting and division in a nonlinear cell population model
Romain Yvinec
To cite this version:
Romain Yvinec. Bursting and division in a nonlinear cell population model. Stochastic Biology: from Cells to Populations, May 2014, Vienne, Austria. 1 p., 2014. �hal-02796022�
Bursting and Division in a Nonlinear Cell Population Model
Romain YVINEC
Inra Nouzilly UMR85 (Tours, France)
MAThematics Center Heidelberg (University of Heidelberg, Germany) romain.yvinec@tours.inra.fr
Summary
We use Pure-Jump Markov processes to describe the stochastic protein bursting production and molecular repartition at division. We find asymptotic convergence criteria and analytical solutions for the steady-state probability density of proteins in a single cell. We also find analytical solutions of mean waiting time to reach a given level. This findings are used to caracterise the behavior of the model as a function of parameters (bifurcation) and can be applied for inverse problems. Finally, we study a population model based on a non-linear extension of the single cell model. We find with numerical simulations situations where a Hopf bifurcation occurs, and where unfrequent but large burst prevents oscillations.
1. The model: Pure-Jump Markov process
The building blocks of this model are two non local operators that represent respectivly the bursting and division.
• Bursting: at rate λb(x), a cell increases its molecular con- tent, from x to y according to the bursting kernel distribution κb(y, x)1{y>x}dy
• Division: at rate λd(x), a cell gives rise to two cells of lower molecular content, y and x−y, according to the (symmetric) division kernel distribution κd(y, x)1{y<x}dy
Following a single cell line, this model gives a one- dimensional pure-jump Markov (X(t))t≥0 on R∗+, whose typical trajectories are shown in figure 1.
0 10 20 30 40 50 60
0 5 10 15 20 25
X(t)
time
One stochastic trajectory, λ
b=10
0 10 20 30 40 50
0 5 10 15 20 25
X(t)
time
One stochastic trajectory, λ
b=100
Figure 1: Single cell sample path trajectories.
Following the whole population, this model gives a measure-valued pure-jump Markov process, that can be rep- resented as a tree (Figure 2)
A cell tree, λ
b=10 A cell tree, λ
b=100
Figure 2: Cell population evolution
2. Single cell model
Following a single cell line, the generator of (X(t))t≥0 is given by (for bounded functions f)
Af(x) = λb(x)
Z ∞
x
(f(y) − f(x))κb(y, x)dy
+ λd(x)
Z x
0
(f(y) − f(x))κd(y, x)dy
.
The evolution equation (Master equation) on the probability density (R
u(t, x)dx = 1) is given by.
∂u(t, x)
∂t = −λb(x)u(t, x) +
Z x
0
λb(y)u(t, y)κb(x, y)dy
− λd(x)u(t, x) +
Z ∞
x
λd(y)u(t, y)κd(x, y)dy
This defines a semi-group P (t) on L1. We will use the Lemma 1. (taken from [4])
If P(t)
• is a stochastic semigroup: kP(t)uk1 = kuk1,
• is partially integral: there exists t0 > 0 and p s.t.
Z ∞
0
Z ∞
0
p(x, y)dy dx > 0 and P(t0)u(x) ≥
Z ∞
0
p(x, y)u(y)dy
• and possess a unique invariant density, then P (t) is asymptotically stable.
2.1 Asymptotic stability of the density
The Master equation may be rewritten as
du
dt = −λu + K(λu), where λ(y) := λb(y) + λd(y) and
Kv(x) :=
Z x
0
λb(y)
λ(y) u(t, y)κb(x, y)dy +
Z ∞
x
λd(y)
λ(y) u(t, y)κd(x, y)dy If K has a strictly positive fixed point in L1, then P (t) is stochas- tic ([5, 1]). We consider the separable kernel case
κb(x, y) = −Kb0(x)
Kb(y), x > y, κd(x, y) = Kd0(x)
Kd(y), x < y.
where Kb(y) → 0 as y → ∞ and K(y) → 0 as y → 0. We define
G(x) = Kd0(x)
Kd(x) − Kb0(x)
Kb(x), Qb(x) =
Z x x
λb(y)
λ(y) G(y)dy.
Theorem 1. Asymptotic stability Suppose that
cb :=
Z ∞
0
Kb(x)
λ(x) G(x)e−Qb(x)dx < ∞,
Z ∞
0
Kb(x)G(x)e−Qb(x)dx < ∞ Then the semigroup {P (t)}t≥0 is stochastic and is asymp-
totically stable. For any initial density u0, u(t, x) converges to
u∗(x) = 1 cb
Kb(x)
λ(x) G(x)e−Qb(x)
Remark 1 Lyapounov-fonction strategy ([3]) can be used to find sufficient conditions of ergodicity in more general cases.
Corollary 1. Bifurcation (see [2])
The number of modes of the stationary solution are linked to the number of solutions of
0 = −λ0(x)
λ(x) + Kb0(x)
Kb(x) + G0(x)
G(x) + λb(x)
λ(x) G(x)
10-3 10-2 10-1 100
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
Figure 3: Kb(x) = e−x/b, λb(x) = λbΛ+x1+xnn, Kd(x) = x.
2.2 Mean waiting time
We can also solve (analytically) the backward equation, Af(x) = A(x). We found for instance that the mean waiting time is non-monotonic with respect to the bursting property.
100 101 102 103 104 105 106 107 108
10-2 10-1 100 101 102 103
mean time
λb
Mean waiting time to go up from x=1 to z=10
b λb =1 b λb =2 b λb =3 b λb =4 b λb =5 b λb =10
10-1 100 101
10-2 10-1 100 101 102 103
mean time
λb
Mean waiting time to go down from z=10 to x=1
b λb =1 b λb =2 b λb =3 b λb =4 b λb =5 b λb =10
Figure 4: Kb(x) = e−x/b, λb(x) ≡ λb, Kd(x) = x.
3. Nonlinear population model
We wish to investigate the (macroscopic) population model with nonlinear feedback on the division rate
∂u(t, x)
∂t = −λb(x)u(t, x) +
Z x
0
λb(y)u(t, y)κb(x, y)dy
− λd(x, S)u(t, x) + 2
Z ∞
x
λd(y, S)u(t, y)κd(x, y)dy − µ(x)u(t, x) where the feeback strenght is given by
S(t) =
Z ∞
0
ψ(x)u(t, x)dx, ψ(x) = 1{x≥x
0}.
We will restrict to the case of constant division and death rates, so that
d dt
Z ∞
0
u(t, x)dx
= (λd(S) − µ)
Z ∞
0
u(t, x)dx
3.1 All cells participate to the feedback
If x0 = 0, S(t) = R ∞
0 u(t, x)dx, and we have immediately Theorem 2. Asymptotic stability
Under the hypothesis of Theorem 1, and if S 7→ λd(S) is continuous monotonically decreasing, with λd(0) > µ and limS→∞ λd(S) < µ, then for any initial density u0, the solu- tion u(t, x) converges as t → ∞ in L1 towards
λ−1d (µ)u∗.
3.2 A fraction on cells participate to the feed- back
In the case x0 > 0, we can only prove a persistance result for the equation
∂u(t, x)
∂t + ∂g(x)u(t, x)
∂x =
− λd(S)u(t, x) + 2
Z ∞
x
λd(S)u(t, y)κd(x, y)dy − µu(t, x)
Theorem 3. Persistance
With g smooth, bounded and bounded away from 0, starting with a positive u0 ∈ L1, we have
0 < inf
t≥0
Z ∞
0
u(t, x)dx ≤ sup
t≥0
Z ∞
0
u(t, x)dx < ∞ 0 < inf
t≥0 S(t) ≤ sup
t≥0
S(t) < ∞
3.3 Numerical results indicate a Hopf bifurca- tion
0 0.5 1 1.5 2 2.5 3
0 2 4 6 8 10
density
x
time evolution of the normalized profile
0 20 40 60 80 100 120 140 160 180
0 200 400 600 800 1000
Number of cells
time
time evolution of the S
value of S(t) candidate limit of S
Figure 5: λd(x, S) ≡ 1+0.1∗S10 ,x0 = 1, g(x) ≡ 0.6
0 0.5 1 1.5 2 2.5 3 3.5
0 2 4 6 8 10
density
x
time evolution of the normalized profile
0 200 400 600 800 1000 1200 1400 1600 1800
0 200 400 600 800 1000
Number of cells
time
time evolution of S
value of S(t) candidate limit of S
Figure 6: λd(x, S) ≡ 1+0.1∗S10 , x0 = 1, g(x) ≡ 0.5
We found that the bursting and the asymmetry of the divi- sion shift the Hopf bifurcation
bλb\λb 100 10 1 0.1
0.6 + + + +
0.5 - + + +
0.4 - - + +
0.1 - - - +
g\p 0.5 0.4 0.2 0.1 0.01
0.7 - + + + +
0.6 - - + + +
0.5 - - - - +
0.4 - - - - -
Table 1: Left: Unfrequent but large burst prevent oscillations.
Right: with κd(·, x) = 0.5N(xp, xp(1 − p)) + 0.5N(x(1 − p), xp(1 − p)), the asymmetry of the division prevents oscillations.
4. Conclusion and Perspectives
Upon an assumption of separable bursting and division kernel, we found a complete characterisation of the single cell model:
• Criteria for convergence towards steady-state, and analytical solution (and bifurcation)
• Mean waiting time to reach a given level
Such study can be used to infer the burst rate and/or division rate in a dividing cell population.
While looking at the nonlinear population model, the bursting properties and division mechanism are shown to have a pro- found impact on homeostasis that will be further investigated.
References
[1] M. C. Mackey, M. Tyran-Kami ´nska, and R. Yvinec. Dynamic behavior of stochastic gene expression models in the presence of bursting. SIAM Journal on Applied Mathematics, 73(5):1830–1852, 2013.
[2] Michael C. Mackey, Marta Tyran-Kami ´nska, and Romain Yvinec. Molecular distributions in gene regulatory dynamics. Journal of Theoretical Biology, 274(1):84–96, April 2011.
[3] S. P. Meyn and R. L. Tweedie. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London Ltd., London, 1993.
[4] Katarzyna Pich ´or and Ryszard Rudnicki. Continuous Markov semigroups and stability of transport equations. J. Math. Anal. Appl., 249(2):668–685, 2000.
[5] Marta Tyran-Kami ´nska. Substochastic semigroups and densities of piecewise deter- ministic Markov processes. J. Math. Anal. Appl., 357(2):385–402, 2009.
