Paweł ŻukWitold BednorzJoanna JaruszewiczTomasz Lipniacki

Noise defines the global attractors in stochastic dynamical systems

Institute of Fundamental Technological Research, Warsaw, Poland

One dimensional Birth-Death process with bistability

UU

KDKK

UU

KBKK

:1

:1

xUK

dx

xdVxWxDxB

dt

dx )(:)(:)()(

Deterministic limit: Stochastic model:

K – number of molecules

U – volume of the reactor

x – substrate concentration

2

2

0 z

xd

dx

dV

t

x

Meaning of the potential V(x)

Travelling wave solutions

dddx

xVxVc 2

31 )()(

)()( xctzxx

Stationary probability distribution

01

1

U

KDF

U

KBF KK

K

iK

Ui

D

Ui

BFF

00 1

In steady state the net probability flow is zero

Thus

K

iK

Ui

D

Ui

BFF

00 1

loglog

x

K

dzzD

zBx

where

xUFxFF

0

0

)(

)(log)(

))(exp()(

For large U

ofglobalxwhere

xFUFor K

min

)(,

Two potentials: V(x) and Φ(x)

dxxBxDxV )()()(

Deterministic „Stochastic”

The minima and maxima of coincide

BUT the global minimum of deterministic potential may not coincide with the global minimum of „stochastic potential”

)()( xandxV

x

dzzB

zDx

0 )(

)(log)(

Thermodynamic interpretation

)/)(( yVyFJF TkD B/

)/( yFDJD

dy

dFyTk

dy

dVF B )(

y

B

dyyTk

dyydVFyF

0

0 ])(

/)(exp[)(

Drift current where - mobility

Diffusion current

))(/)(log(

)()(1)(

xDxB

xDxB

UkxT

B

In the steady state DF JJ

SPD of B-D process is proportional to density of particles diffusing in potential and temperature fields

dxxBxDxV )()()(

For any potential

having the minima in there exists

Such that

dxxBxDxV )()()(

31 xandx

x

dzxfzB

xfzDx

0 )()(

)()(log)(

has global minimum in 1x

0)( xf

REMARK

That is, any of macroscopic stable steady states may became a global attractor

depending on noise or temprature profile.

Example: bistable kinase activation model

fbxxD

fxxccxB

)(

)1)(()( 22

f - the flow of active kinase to and from the compartment

We focus on the case in which

has three roots

)()()( xDxBxW

321, xandxx

1321 xxx

Bistability domain

and separatrices

Bimodal SPD expected for

bistable systems

may be observed

only if the magnitude of noise

is sufficiently large

For symmetric potential V(x)

SPD concentrates in colder

attraction basin.

For f = 0 the temperature profile is not uniform and for the symmetric potential V(x) SPD concentrates in the colder potential attraction basin of point x1. The temperature effect is balanced by asymmetry of the potential.

For f=1 the temperature profile is flatter and SPD concetrates in the deeper potential well

The flow controls temperature profile T(x)and SPD F(x).

Bistable model of the autoregulatory gene

Bistable model of the autoregulatory gene

1

1

01

10

nYnY

nYnY

SS

SS

n

QS

Yb

Yc

)(

)(

nnnnn

nnnnnnn

gnbhncnhhndt

dh

gnbhncnggnggQdt

dg

)()()1(

)()()1()(

1

11

0,,

)(

)/()(

020

0

2220

bcc

bnb

nQccnc

The stochastic model and its three approximations

Continuous large number of proteins

Adiabaticfast gene transitions

Deterministic approximation

yySdt

dy )(

QYy /

0)()( 0002

23

2 cybcycycyW

)1()( 132321

32100 yyyyyy

yyybc

)1()( 132321

02 yyyyyy

bc

1321 yyy

ODE for protein concentration

the expected value of gene state )()(

)()(

ybyc

ycyS

Steady state solutions satisfy0

220 )(,)( bybyccyc

Roots of W(y) satisfy

Continuous approximation

ySdt

dy

QYy /

01

10

SS

SS

)(

)(

yb

yc

gychybhyyt

h

gychybygyt

g

)()())1((

)()()(

y

yygyh

dss

yc

s

ybCExpyg

y

1

)()(

])1)(

1

)(([)(

0

112

1

00200

22

)1()()(),,;( bcycyyCeyhygbccyf

protein concentration

PDE for probability density functions g(y,t) and h(y,t)

Separatrix in the continuous approximation

))(1(2

12

))(1)(1(

)1(

1)1)(

1(

211

212

2121

21211

21

1

21

1 21

yyy

yyp

yyyy

yyyyp

eyy

y

yy

y pp

0)),(),(;(

)),(),(;(

000023

000021lim0

constbbcbcyf

bbcbcyf

b

Simulation of the continuous model

0/1 b gene switching noise parametr

7.0

27.0

03.0

3

2

1

y

y

y

Adiabatic approximation

nnD

nQSnB

)(

)()(

)()(

)()(

nbnc

ncnS

0)1()( 1 nDFnBF nn

1

10 )1(

)(n

in iD

iBFF

Birth and death process with transition rates

where

In steady state the net probability current equals to zero

Stationary probability distribution

Simulation of the adiabatic approximation

Q/1 transcriptional noise parameter

7.0

27.0

03.0

3

2

1

y

y

y

Adiabatic approximation (zero noise limit)

1

1 )/1/(

)/()0()(yQ

i

QQ

QQid

QibQFyF

nQ QFyF )( Q

ny

QiDQid

QiBQib

)(:)/(

)(:)/(

1

1 )/1/(

)/(log)0(loglog)(log

yQ

i

QQ

QQid

QibFQyF

y

Q dzzd

zbQFQ0 )(

)(log)0(loglog

0

))(exp(

))(exp()(

dzyQ

yQyF Q

y

dzzd

zby

0 )(

)(log)(

Let where

where

where

))(exp()(

2)0())(exp()0()( m

m

Qyy

yy

yy

yy

QQ yQyQ

QFdzzQQFdzzFm

m

m

m

Qyyd

byccyccQyb

)(

)/()()( 02

202

20

]))((

log[1(

])(

arctan[)(2

]arctan[2)(

232132

2321

32132

32132

321

321

yyyyyyk

yyyyy

yyyyy

yyyyyy

yyy

yyyyy

For

Separatrix for the adiabatic approximation

)()( 31 yy

)1(3

1lim constF

FQy

Qy

Q

12

])1()1(

1arctan[)1()1(2

])1()1(

arctan[)1()1(2

])1(

1arctan[)1(2

])1(

arctan[)1(20

21

11212

2111212

11212

111212

2121

212121

2121

12121

yy

yyyyy

yyyyyyy

yyyyy

yyyyyy

yyyy

yyyyyy

yyyy

yyyyy

Bistability domain: separatrices for continuous and adiabatic

approximations

PDF for the adiabatic and continuous approximation

PDF versus noise ratio parametr /