Towards nonlinear cell population model structured by molecular content

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Towards nonlinear cell population model structured by molecular content

Romain Yvinec, Michael C. Mackey, M. Tyran-Kamińska, A.

Marciniak-Czochra

To cite this version:

Romain Yvinec, Michael C. Mackey, M. Tyran-Kamińska, A. Marciniak-Czochra. Towards nonlinear

cell population model structured by molecular content. Structured Integro-Differential models in

Mathematical Biology, Vienna International Science Conferences and Events Association. AUT., Apr

2014, Vienne, Austria. �hal-02797854�

Single cell Population model

Towards nonlinear cell population model structured by molecular content

Romain Yvinec

Inra Tours - Match Heidelberg

Joint work with M. Mackey, M. Tyran-Kaminska, A. Marciniak-Czochra

0/30

Outline

Bursting and Division in gene expression models Stochasticity in Molecular biology

Bursting and Division as Jump Processes

Nonlinear population model

Theoretical results

Numerical results

Outline

Bursting and Division in gene expression models Stochasticity in Molecular biology

Bursting and Division as Jump Processes

Nonlinear population model

Theoretical results

Numerical results

Stochasticity in molecular biology

[Eldar and Elowitz Nature 2010]

Single cell Population model Biology Jump process

Much more accurate measurements

I The bursting event are well characterized

[Yu et al. Science 06]

2/30

Much more accurate measurements

I Trajectories can be analyzed on single cells.

[Golding et al. Cell 2005, Kondev Physics Today 2014]

Single cell Population model Biology Jump process

Much more accurate measurements

I Bifurcation can be studied on probability distributions.

[Song et al. Plos CB 2010, Mackey et al. JTB 2011, SIAM 2013]

4/30

A typical example linking gene expression to cell fate

The antagonism Gata-1/PU.1 in heamatopoietic progenitor

[Enver et al. Stem Cell 2009]

Single cell Population model Biology Jump process

A typical example linking gene expression to cell fate

The antagonism Gata-1/PU1, modeled by ODE

[Duff et al. JMB 2012]

6/30

A typical example linking gene expression to cell fate

[Chang et al. Nature Letters 08] [Pina et al. Nature cell bio. 2012]

Outline

Bursting and Division in gene expression models Stochasticity in Molecular biology

Bursting and Division as Jump Processes

Nonlinear population model

Theoretical results

Numerical results

We define a pure-jump process (X (t)) t≥0 on R ^∗ ₊ with two different transitions :

I Bursting at rate λ b (x) and jump distribution κ b (y, x)1 {y>x} dy

I Division at rate λ _d (x) and jump distribution κ _d (y, x)1 _{y<x} dy Pathwise construction with the sequence (U _n , V _n ) n≥1 , of i.i.d uniform random variable on (0, 1)

I T _n = T n−1 − (1/λ(X n−1 )) ln(U n−1 ), where λ(x) = λ _b (x) + λ _d (x).

I X n = F _K ⁻¹ (V n , X n−1 ), where F _K (y, x) is the cum. dist. fonct.

associated to K (y , x ) = λ

(x)

λ

(x ) + λ

(x) κ

(y, x)1

+ λ

(x)

λ

(x) + λ

(x) κ

(y, x)1

.

I X (t) = X n−1 for all T n−1 ≤ t < T _n .

Single cell Population model Biology Jump process

This model is well-defined up to the explosion time, T ∞ = lim

n→∞ T n

A well-known sufficient condition for non-explosion (T ∞ = ∞) is given by

X

n≥0

1

λ _b (X _n ) + λ _d (X _n ) = ∞.

In particular, this is the case for bounded jump rate.

9/30

Another criteria is provided by Lyapounov-fonction strategy (see [Meyn and Tweedie 93]). Let A be the generator of (X (t)) t≥0 ,

Af (x) = λ _b (x) Z ∞ x

(f (y ) − f (x))κ _b (y, x)dy + λ _d (x)

Z x

0

(f (y ) − f (x))κ _d (y, x)dy

. If there exists c > 0, V a positive measurable function s.t V (x) → ∞ when x → 0 and x → ∞, V ∈ D(A) and

AV (x) ≤ cV (x ), x > 0,

then (X (t)) t≥0 is non-explosif.

Single cell Population model Biology Jump process

The fonction V (x) = x ^−γ 1 _{x≤1} + x ^α 1 _{x>1} is suitable if there exists A, B , β, δ,

I κ _b (y, x) = R ∞

y κ _b (z , x)dz ≤ c (x/y ) ^β , β > α

I κ d (y , x) = R y

0 κ d (z , x)dz ≤ c (y/x) ^δ , δ > γ

I λ _d (x) < Aλ _b (x) + B as x → 0 and

x→0 lim λ b (x)x ^δ Z 1

x

y ^−δ κ b (y , x)dy < ∞

I λ _b (x) < Aλ _d (x) + B as x → ∞ and

x→∞ lim λ _d (x)x ^−α Z x

1

y ^α κ _d (y, x)dy < ∞ Remark

“Similar” condition holds for ergodicity.

Remark

Non-explosion + irreductibility + Existence of a unique invariant measure ⇒ ergodicity.

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I An analogous study on the set of probability density ( R

u = 1).

∂u(t , x )

∂t = −λ

(x)u(t , x ) + Z

0

λ

(y )u(t, y)κ

(x , y )dy

− λ

(x)u(t , x) + Z

x

λ

(y)u(t , y )κ

(x , y )dy This defines a semi-group P(t) on L ¹ . We will use

Theorem (Pichor and Rudnicki JM2A 2000) If P(t)

I is a stochastic semigroup : kP (t)uk ₁ = kuk ₁ ,

I is partially integral : there exists t ₀ > 0 and p s.t.

Z

0

Z

0

p(x, y ) dy dx > 0 and P(t

)u(x) ≥ Z

0

p(x, y)u(y ) dy

I and possess a unique invariant density,

then P(t) is asymptotically stable.

Single cell Population model Biology Jump process

The Master equation may be rewritten as du

dt = −λu + K (λu), (1)

where Kv (x ) =

Z _x

0

λ _b (y)

λ _b (y) + λ _d (y) u(t, y)κ _b (x, y)dy +

Z ∞

x

λ d (y )

λ _b (y) + λ _d (y) u(t, y)κ d (x, y )dy If K has a strictly positive fixed point in L ¹ , then P (t) is stochastic ([Mackey et al. SIAM 13]). Note also that any stationary solution u ^∗ of (1) must satisfy the flux condition

Z x

0

κ _b (x, y)λ _b (y)u ^∗ (y)dy = Z ∞

x

κ _d (x, y )λ _d (y )u ^∗ (y)dy

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We consider the separable case κ

(x, y ) = − K

(x)

K

(y ) , x > y, κ

(x, y) = K

(x)

K

(y ) , x < y .

where K _b (y) → 0 as y → ∞ and K (y ) → 0 as y → 0. We define G (x ) = K

(x)

K

(x) − K

(x)

K

(x) , Q

(x) = Z

x

λ

(y )

λ(y) G (y)dy.

Theorem Suppose that c _b :=

Z ∞

0

K _b (x)

λ(x) G (x)e ^−Q

^(x) dx < ∞, Z ∞

0

K _b (x)G (x)e ^−Q

^(x) dx < ∞ Then the semigroup {P (t)} _t≥0 is stochastic and is asymptotically stable, with

u ∗ (x) = 1 c _b

K _b (x)

λ(x) G (x )e ^−Q

^(x)

Single cell Population model Biology Jump process

du ^∗ dx = h

− λ ⁰ (x)

λ(x) + K _b ⁰ (x)

K _b (x) + G ⁰ (x)

G (x) + λ _b (x) λ(x) G (x) i

u ^∗ (x)

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

K _b (x) = e ^−x/b , λ _b (x) = λ _{b Λ+x} ^1+x

, K _d (x) = x, λ _d (x) = 1.

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I This theorem can be used to show asymptotic convergence for

“non-trivial” parameters function.

In particular, the growth-division model

∂u(t , x)

∂t + ∂g (x)u(t, x)

∂x = −λ

(x )u(t, x) + Z

x

λ

(y )u(t, y) K

(x) K

(y ) dy, converges for

λ d (x) = αx ^β−1 + x ^β+1 g (x) = x ^β

K d (x) = x, for 0 ≤ β ≤ 1, 0 < α < 1, towards

u ∗ (x) = K d (x) cg (x) e ⁻

R

¯ x

λd(y) g(y)

dy

,

but λ d

g ∈ / L ¹ ₀

Single cell Population model Biology Jump process

Absorbing probabilities/ Mean waiting time : We can also solve (analytically) the backward equation, Af (x) = A(x).

If

τ _u,z := inf {t ≥ 0, X _t ≥ z }, then

V _u,z (y ) = E y

τ _u,z is solution of

( AV _u,z (y) = −1, y < z,

V u,z (y) = 0, y ≥ z. (2)

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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

λ b=1, b=10 λ

b=5, b=2 λ

b=10, b=1 λ b=100, b=0.1

0 1 2 3 4 5 6 7

0 2 4 6 8 10 12 14

mean time

x

Mean waiting time to go up from x to z=10

λb=1, b=10 λb=5, b=2 λb=10, b=1 λb=100, b=0.1

0 5 10 15 20

0 5 10 15 20 25 30

mean time

z

Mean waiting time to go down from z to x=0.5

λb=1, b=10 λb=5, b=2 λb=10, b=1 λb=100, b=0.1

K _b (x) = e ^−x/b , λ _b (x) = λ _{b Λ+x} ^1+x

, K _d (x) = x, λ _d (x) ≡ 1.

Single cell Population model Biology Jump process

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

λ b=1, b=10 λ

b=5, b=2 λ

b=10, b=1 λ b=100, b=0.1

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

K _b (x) = e ^−x/b , λ _b (x) = λ _{b Λ+x} ^1+x

, K _d (x) = x, λ _d (x) ≡ 1.

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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

λ_b

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

λ_b

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

λ d ≡ 2, K d (x) = x, λ b (x) ≡ λ b , K b (x) = e ^−x/b

Single cell Population model Biology Jump process

The mean waiting time is also affected by the asymmetry of the division.

10¹ 10² 10³

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

mean time

p

Mean waiting time to go up from x=1 to z=10

λb=1 λb=10

10^-1 10⁰ 10¹ 10²

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

mean time

p

Mean waiting time to go down from z=10 to x=1

λb=1 λb=10

λ _d (x) ≡ 2,

K d (x) = 0.5N (xp, xp(1 − p)) + 0.5N (x(1 − p ), xp(1 − p)), K _b (x) = e ^−x/b , bλ _b = 2

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Outline

Bursting and Division in gene expression models Stochasticity in Molecular biology

Bursting and Division as Jump Processes

Nonlinear population model

Theoretical results

Numerical results

Single cell Population model Theory Numerics

We wish to investigate (macroscopic) population models with nonlinear feedback on the division rate

∂u(t , x )

∂t = −λ

(x)u(t , x ) + Z

0

λ

(y )u(t, y)κ

(x , y )dy

− λ

(x, S)u(t , x ) + 2 Z

x

λ

(y, S )u(t, y)κ

(x, y)dy − µ(x )u(t, x) with κ _d symmetric (total molecular content preserved at division) the feeback strenght is given by

S (t) = Z ∞

0

ψ(x)u(t, x)dx , ψ(x) = 1 _{x≥x

_} .

We will restrict to the case of constant division and death rates, so

that d

dt Z ∞

0

u(t, x)dx

= (λ(S ) − µ) Z ∞

0

u(t, x)dx

22/30

If all cells participate to the regulation of the division rate (x 0 = 0), we have immediately

Theorem

Let κ b (x, y) = − ^K _K

^(x)

b

(y ) , and κ d (x, y) = ^K

0 d

(x)

K

(y) . We assume c _b :=

Z ∞

0

K b (x)

λ(x) G (x)e ^−Q

^(x) dx < ∞, Z ∞

0

K _b (x)G (x)e ^−Q

^(x) dx < ∞ and that S 7→ λ _d (S) is continuous monotonically decreasing, with λ d (0) > µ and lim S →∞ λ d (S) < µ, then, for any initial density u 0 , u(t, x) converges as t → ∞ in L ¹ towards

λ ⁻¹ _d (µ)u ^∗ .

Single cell Population model Theory Numerics

In the case x ₀ > 0, we can only prove a persistance result for the equation

∂u(t , x )

∂t + ∂g (x)u(t , x)

∂x =

− λ

(S)u(t , x) + 2 Z

x

λ

(S )u(t , y)κ

(x, y)dy − µu(t, x)

Theorem

With g smooth, bounded and bounded away from 0, starting with a positive u 0 ∈ L ¹ , 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) < ∞

24/30

D´ emonstration.

We define v(t, x) := e ^R

^(µ−λ

^(S(s)))ds u(t, x), so that

∂v (t , x)

∂t + ∂g (x )v (t, x)

∂x = −2λ

(S)v (t, x)+2λ

(S) Z

x

v(t , y )κ

(x, y )dy

We use a coupling strategy to show that Z ∞

x

v(t, x)dx ≥ c (1 + ε(t ))

with ε(t) → 0 (at exponential speed). For this, we use the coupling Af (x, y) = g (x)f ⁰ (x) + g (y)f ⁰ (y)

+ 2λ d (S (t )) Z 1

0

(f (xz , yz) − f (x, y ))dz

+ 2(kλ _d k _∞ − λ _d (S(t))) Z ₁

0

(f (xz, y) − f (x, y ))dz

.

Single cell Population model Theory Numerics

Then, R ∞

x

v(t, x)dx ≥ R ∞

x

w (t, x)dx where

∂w (t, x)

∂t + ∂g (x)w (t , x)

∂x = −2kλ

k

w (t, x)+2kλ

k

Z

x

w (t , y )κ

(x , y )dy which converges as t → ∞ due to hypotheses on g , κ _d .

26/30

Outline

Bursting and Division in gene expression models Stochasticity in Molecular biology

Bursting and Division as Jump Processes

Nonlinear population model

Theoretical results

Numerical results

Single cell Population model Theory Numerics

Numerical results

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

µ = 1, λ d (x, S ) ≡ _1+0.1∗S ¹⁰ , K d (x) = x, x 0 = 1, g (x) ≡ 0.6

27/30

Numerical results indicate a Hopf bifurcation

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

µ = 1, λ _d (x, S ) ≡ _1+0.1∗S ¹⁰ , K _d (x) = x, x ₀ = 1, g (x) ≡ 0.5

Single cell Population model Theory Numerics

The bursting property shifts the Hopf bifurcation : with µ = 1, λ _d (x, S ) ≡ _1+0.1∗S ¹⁰ , K _d (x) = x, x ₀ = 1, K _b (x) = e ^−x/b , λ b (x) ≡ λ b

bλ b \λ _b 100 10 1 0.1

0.6 + + + +

0.5 - + + +

0.4 - - + +

0.1 - - - +

Table : +=Asymptotic convergence towards steady state - = oscillation

29/30

The asymmetry at division also shifts the Hopf bifurcation : with µ = 1, λ _d (x, S ) ≡ _1+0.1∗S ¹⁰ ,

κ _d (·, x) = 0.5N (xp, xp(1 − p)) + 0.5N (x(1 − p), xp(1 − p)), x 0 = 1, g (x) ≡ g

g \p 0.5 0.4 0.2 0.1 0.01

0.7 - + + + +

0.6 - - + + +

0.5 - - - - +

Table : +=Asymptotic convergence towards steady state - = oscillation

Single cell Population model Theory Numerics

Vielen Dank !

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