Towards nonlinear cell population model structured by molecular content

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Towards nonlinear cell population model structured by molecular content

Romain Yvinec, M. Mackey, M. Tyran-Kamińska, A. Marciniak-Czochra

To cite this version:

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

population model structured by molecular content. 9. European Conference on Mathematical and

Theoretical Biology - ECMTB14, Chalmers University of Technology. SWE., Jun 2014, Göteborg,

Sweden. �hal-02797850�

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

Outline

Stochasticity in Molecular biology, links with cell fate

Single cell model : Bursting and Division as Jump Processes

Nonlinear (macroscopic) population model Theoretical results

Numerical results

Outline

Stochasticity in Molecular biology, links with cell fate

Single cell model : Bursting and Division as Jump Processes

Nonlinear (macroscopic) population model

Biology Single cell model Population model

Stochasticity in molecular biology

I

Trajectories on single cells : bursting and repartition at division are major sources of randomness in gene expression.

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

1/23

A typical example linking gene expression to cell fate

The antagonism between

regulatory proteins (Transcription Factor) Gata-1/PU.1 in

heamatopoietic progenitor

[Enver et al. Stem Cell 2009]

Biology Single cell model Population model

Cell fate explained by a deterministic dynamical system

The antagonism Gata-1/PU1, modeled by ODE

Cell fate “=” attractor of a dynamical system.

[Duff et al. JMB 2012]

3/23

Reversibility of the gene expresion profile (and cell fate ?) in in vitro cell culture experiment

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

Outline

Stochasticity in Molecular biology, links with cell fate

Single cell model : Bursting and Division as Jump Processes

Nonlinear (macroscopic) population model

We define a pure-jump process (X (t))

on R

with two different transitions :

I

Bursting at rate λ

(x) and jump distribution κ

(y, x)1

dy

I

Division at rate λ

(x) and jump distribution κ

(y, x)1

dy Pathwise construction : Let (U

, V

)

be i.i.d ∝ U (0, 1),

I

Time step : T

= T

+ (1/λ(X

)) ln(1/U

), where λ(x) = λ

(x) + λ

(x).

I

State step : X

= F

(V

, X

), where F

(y, x) is the cum.

dist. fonct. associated to

κ(y, x) = λ

(x)

λ

(x) + λ

(x) κ

(y , x )1

| {z }

Bursting (gain)

+ λ

(x)

λ

(x) + λ

(x ) κ

(y , x )1

| {z }

Division (loss)

.

I

X (t) = X

for all T

≤ t < T

.

Biology Single cell model Population model

Example of sample paths

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

6/23

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

= lim

n→∞

T

Remark

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

Lyapounov-fonction strategy (see [Meyn and Tweedie 93]) can

provide sufficient criteria.

Biology Single cell model Population model

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

| {z }

Bursting (gain)

−λ

(x)u(t , x) + Z

x

λ

(y)u(t , y )κ

(x , y )dy

| {z }

Division (loss)

This defines a semi-group P(t) on L

. We will use Theorem (Pichor and Rudnicki JM2A 2000)

If P(t) is a stochastic semigroup : kP (t)uk

= ku k

, is partially integral, e.g. there exists t

> 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 and if P(t) possess a unique invariant density, then P(t) is asymptotically stable.

8/23

The Master equation may be rewritten as du

dt = −λu + K (λu), (1)

where Kv (x ) =

Z

λ

(y)

λ

(y) + λ

(y) u(t, y)κ

(x, y)dy

| {z }

Bursting (gain)

+ Z

x

λ

(y )

λ

(y) + λ

(y) u(t, y)κ

(x, y )dy

| {z }

Division (loss)

If K has a strictly positive fixed point in L

, 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

0

Z

x

κ

(z, y)dz

λ

(y )u

(y )dy

| {z }

“ fromx⁻tox⁺”

= Z

x

Z

0

κ

(z , y)dz

λ

(y)u

(y)dy

| {z }

“ fromx⁺tox⁻”

Biology Single cell model Population model

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

(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

:=

Z

K

(x)

λ(x) G (x)e

dx < ∞, Z

0

K

(x)G (x)e

dx < ∞ Then the semigroup {P (t)}

is stochastic and is asymptotically stable, with

u

(x) = 1 c

K

(x)

λ(x) G (x )e

10/23

du

dx = h

− λ

(x)

λ(x) + K

(x)

K

(x) + G

(x)

G (x) + λ

(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

(x) = e

, λ

(x) = λ

Λ+xⁿ

, K

(x) = x, λ

(x) = 1.

Biology Single cell model Population model

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

| {z }

Continuous production

= −λ

(x )u(t, x) + Z

x

λ

(y )u(t, y) K

(x) K

(y) dy

| {z }

Division (loss)

,

converges for

λ

(x) = αx

+ x

g (x) = x

K

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

u

(x) = K

(x) cg (x) e

Rx

¯ x

λd(y) g(y)dy

,

but λ

g ∈ / L

12/23

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

Af (x) = λ

(x) Z

x

(f (y ) − f (x))κ

(y, x)dy

| {z }

Bursting (gain)

+ λ

(x) Z

(f (y ) − f (x))κ

(y, x)dy

| {z }

Division (loss)

.

If

τ

:= inf{t ≥ 0, X

≥ z }, then

V

(y) = E

τ

is solution of (

AV

(y ) = −1, y < z ,

V

(y ) = 0, y ≥ z. (2)

Biology Single cell model Population model

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

≡ 2, K

(x) = x, λ

(x) ≡ λ

, K

(x) = e

14/23

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

(x) ≡ 2,

κ

(·, x) = 0.5N (xp, xp(1 − p )) + 0.5N (x(1 − p), xp(1 − p)),

K

(x) = e

, bλ

= 2

Outline

Stochasticity in Molecular biology, links with cell fate Single cell model : Bursting and Division as Jump Processes Nonlinear (macroscopic) population model

Theoretical results

Numerical results

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

| {z }

Bursting

−λ

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

x

λ

(y, S)u(t , y )κ

(x, y )dy

| {z }

Division

− µ(x )u(t, x)

| {z }

Cell death

with κ

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

.

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

Biology Single cell model Population model Theory Numerics

If all cells participate to the regulation of the division rate (x

= 0), we have immediately

Theorem

Let κ

(x, y) = −

b(y)

, and κ

(x, y) =

0 d(x)

Kd(y)

. We assume c

:=

Z

K

(x)

λ(x) G (x)e

dx < ∞, Z

0

K

(x)G (x)e

dx < ∞ and that S 7→ λ

(S) is continuous monotonically decreasing, with λ

(0) > µ and lim

λ

(S) < µ, then, for any initial density u

, u(t, x) converges as t → ∞ in L

towards

λ

(µ)u

.

17/23

In the case x

> 0, we can not prove convergence towards a steady-state, and numerical results indicate the presence of oscillation through a Hopf-bifurcation.

Remark

We can however prove persistance results in certain cases 0 < inf

t≥0

Z

u(t, x )dx ≤ sup

t≥0

Z

u(t, x)dx < ∞

Biology Single cell model Population model Theory Numerics

Numerical results

∂u(t,x)

∂t +∂g(x)u(t,x)

∂x

| {z }

Continuous production

=−λ_d(x,S)u(t,x) +2 Z_∞

x

λ_d(y,S)u(t,y)κ_d(x,y)dy

| {z }

Division

−µ(x)u(t,x)

| {z }

Cell death

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

(x, S ) ≡

, K

(x) = x, x

= 1, g (x) ≡ 0.6

19/23

Numerical results indicate a Hopf bifurcation

∂u(t,x)

∂t +∂g(x)u(t,x)

∂x

| {z }

Continuous production

=−λ_d(x,S)u(t,x) +2 Z_∞

x

λ_d(y,S)u(t,y)κ_d(x,y)dy

| {z }

Division

−µ(x)u(t,x)

| {z }

Cell death

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

(x, S ) ≡

, K

(x) = x, x

= 1, g (x) ≡ 0.5

Biology Single cell model Population model Theory Numerics

The bursting property shifts the Hopf bifurcation

∂u(t,x)

∂t =−λ_b(x)u(t,x) + Z_x

0

λ_b(y)u(t,y)κ_b(x,y)dy

| {z }

Bursting

−λ_d(x,S)u(t,x) +2 Z∞

x

λ_d(y,S)u(t,y)κ_d(x,y)dy

| {z }

Division

−µ(x)u(t,x)

| {z }

Cell death

with µ = 1, λ

(x, S ) ≡

, K

(x) = x, x

= 1, K

(x) = e

, λ

(x) ≡ λ

bλ

\λ

100 10 1 0.1

0.6 + + + +

0.5 - + + +

0.4 - - + +

0.1 - - - +

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

21/23

The asymmetry at division also shifts the Hopf bifurcation

∂u(t,x)

∂t +∂g(x)u(t,x)

∂x

| {z }

Continuous production

=−λ_d(x,S)u(t,x) +2 Z_∞

x

λ_d(y,S)u(t,y)κ_d(x,y)dy

| {z }

Division

−µ(x)u(t,x)

| {z }

Cell death

with µ = 1, λ

(x, S ) ≡

,

κ

(·, x) = 0.5N (xp, xp(1 − p)) + 0.5N (x(1 − p), xp(1 − p)), x

= 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

Biology Single cell model Population model Theory Numerics

Upon an assumption of separable bursting and division kernel, we found a complete characterisation of the single cell model :

I

Criteria for convergence towards steady-state, and analytical solution (and bifurcation)

I

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 profound impact on homeostasis that will be further investigated.

Thank you for your attention !

23/23