Non-Markovian dynamics of small genetic circuits Lev Tsimring Institute for Nonlinear Science University of California, San Diego Le Houches, 9-20 April,

Non-Markovian dynamics of small genetic circuits

Lev TsimringInstitute for Nonlinear Science

University of California, San Diego

Le Houches, 9-20 April, 2007

Outline

• Deterministic and stochastic descriptions of genetic circuits with very different time scales

• Non-Markovian effects in gene regulation– transcriptional delay-induced

stochastic oscillations

Gene regulatory networks• Proteins affect rates of production of other proteins

(or themselves)• This leads to formation of networks of interacting

genes/proteins• Different reaction channels operate at vastly

different time scales and number densities• Sub-networks are non-Markovian, even if the whole

system is • Compound reactions are non-Markovian

A

B

C

D E

A

B

D

Transients in gene regulation

• Genetic circuits are never at a fixed point:– Cell cycle; volume growth; division– External signaling– Intrinsic noise– Extrinsic “noise”– Circadian rhythms; ultradian rhythms

External signaling: -Phage Life Cycle

M.Ptashne, 2002

Engineered Toggle Switch

Gardner, Cantor & Collins, Nature 403:339 (2001)

Construction/experiments:

Model

Gene A onGene B off Reporter

GFPRepressor A

Gene A offGene B on Reporter

Repressor B

“On”

“Off”

Circadian clock in Neurospora crassa

2

2

2

2

( )

1 ( )

( )

1 ( )

ff

w

fw

f

k w tf f fw

K w t

k f tw w fw

K f t

WC-1

WC-2WCC FRQ

P.Ruoff

Ultradian clock at yeastKlevecz et al, 2004

5,329 expressed genes

Reductive phaseR

espi

rato

ry p

hase

Average peak-to-trough ratio ~2

Synchronized culture

The Repressilator

G FPA

B C

Gene A

Gene B Gene C

Elowitz and Leibler, Nature 403:335 (2001)

Model

Construction/experiments:

RNAP

Auto-repressor: A cartoon

promoter gene

RNAP

DNA

Binding/unbinding rate: <1 secTranscription rate: ~103 basepairs/minTranslation rate: ~102 aminoacids/minmRNA degradation rate ~3minTransport in/out nucleus 10+ minProtein degradation rate ~ 30min ..hours protein

mRNA

Oscillations in gene regulation

promoter gene

RNAP

DNA

RNAP

repressor

mRNA

Single gene autoregulation

tk

tk

1k

2k 1k2k

21 XXX k

XXX k 12

1202 DXD k

2012 XDD k

Fast

Slow

Binding/unbinding rate (k-1,k-2): ~1 sec-1

Transcription rate (kt): ~1 min-1..0.01 min-1

Protein degradation (kx) ~0.01 min-1xkX

0 0tkD P D P X

1 1tkD P D P X

Single gene autoregulation

21 XXX k

XXX k 12

1202 DXD k

2012 XDD k

xkX

0 0tkD P D P X

1 1tkD P D P X

Fast

Slow

21 1 2

22 1 1 2 2 0 2 2 1

0 2 0 2 2 1

0

1 2 2

1

2 0

0

1

( )2 2 x tx k x k x

x k x k x k d x k d

d k d x k d

d k d

k x k p d d

x k d

Quasi-steady-state approximation (naïve approach)

xkxKK

xKKdpkx xt

2

21

221

0 1

1

000

Fixed points – yes,Dynamics – no: x is a fast variable also!

2021 xdKd 2

12 xKx ddd 10

Correct?

Separation of scales(Correct projection)

21 2

0 21 2

1 21 2 2

1 2

1

1

4( ) 1 4

(1 )

x t x

t

K K xx k p d k x

K K x

dK K xx x K x

K K x

x

P

P

2021 xdKd 2

12 xKx

ddd 10

000

2 12 2tx x x d slow variable

• Prefactor is important if x2/x~1, i.e. lots of dimers• Prefactor makes transients slower

Kepler & Elston, 2001Bundschuh et al, 2003Bennett et al, 2007, in press

0 0 1( )t t xx k p d d k x

22 1 2

1 21 2

2 21

K K xx K x d

K K x

Genetic toggle switch[Gardner, Cantor, Collins, Nature 2000]

Gene A onGene B off Reporter

GFPProtein A

Gene A offGene B on Reporter

Protein B

“On”

“Off”

Multiple-scale analysis

(0) (1) 2 2

, ,

( ) ( , ) ( , ) ( ); ( )

t t x x

t t T

k k k k T t

z t z t T z t T O d O

(0) (0) 2 (0)1 1 2

(0) (0) 2 (0) (0) (0) (0)2 1 1 2 2 0 2 2 1

(0) (0) (0) (0)0 2 0 2 2 1

(0) (0) (0) (0)1 2 0 2 2

0

1

2 [ ] 2

[ ]

( ) :

t

t

t

x k x k x

x k x k x k d x k d

d k d x k d

d k d d

O

x k

(1) (0) (1) (1) (0) (0) (0) (0)1 1 2 0 0 1

(1) (0) (1) (1) (0) (1) (1) (0) (1) (0)2 1 1 2 2 0 2 0 2 2 1 2

(1) (0) (1) (0) (1) (0) (0)0 2 0 2 0 2 2 1 0

(1)1

1( )

4 2 ( )

2 ( )

(

:

)

t T x t

t T

t T

t

x k x x k x x k x k p d d

x k x x k x k d x d x k d x

d k d x d x k d

d k

O

d

(0) (1) (0) (1) (0) (0)2 0 2 0 2 2 1 1( ) Td x d x k d d

Fast reactions

Multiple-scale analysis (cont’d)

ddd

xdxx t

)0(

1)0(

0

)0(1

)0(2

)0( 22 slow variableconstant

Local equilibrium for fast reactions Nullspace of the adjoint linear operator

[2 eigenvectors]

From orthogonality conditions:

)0(2

)0(02

)0(1

2)0(1

)0(2 ][

xdKd

xKx

}1,1,0,0{

}2,0,2,1{

102

102

ddxx

ddxx

2)0(21

2)0(212)0(

1)0(

][1

][2][2

xKK

xKdKxKxxt

21 2

0 21 2

1 21 2 2

1 2

1

1

4( ) 1 4

(1 )

x t x

t

K K xx k p d k x

K K x

dK K xx x K x

K K x

x

P

P

Prefactor

w/o prefactorwith prefactorfull model

1 21 2 2

1 2

4( ) 1 4 ~ 40

(1 )t

dK K xx x K x

K K x

xP

Stochastic gene expression: Master Equation approach

( )xp t

1 1( ) ( 1)xt x x x x x

dpk p p k x p xp

dt

Two reactions: production degradationt

kX x

kX

Probability of having x molecules of X at time t,

Dynamics of ( ) :xp t

Continuum limit (x>>1): ( ) ( , )xp t p x t2

22

( , ) 1 ( , )( 1, ) ( , ) 1 1 ...

2

f x t f x tf x t f x t

x x

2

2

( , ) 1[( ) ( , )] [( ) ( , )]

2t x t x

p x tk k x p x t k k x p x t

t x x

Fokker-Planck equation

Stochastic gene expression: Langevin equation approach

Two reactions: production degradationt

kX x

kX

Deterministic equation:

2

2

( , ) 1[( ) ( , )] [( ) ( , )]

2t x t x

p x tk k x p x t k k x p x t

t x x

t xx k k x Each reaction is a noisy Poisson process, mean=variance

( )t tx k k t ( )x xx k x k x t Separately:

Since reactions are uncorrelated, variances add:

t x t x tx k k x k k x Langevin equations

From Langevin equation to FPE (van Kampen, Stochastic Processes in Chemistry and Physics,1992):

…or from FPE to Langevin!

Autoregulation: stochastic description

Master equation for bunp ,,

Projection: using nbunbun ppp |,,,

n – total # of monomers; u – # of unbound dimers; b - # of bound dimers

nbnunm

mmKK

mmKdKnbmmKnu

|2|2

)1(1

)1(|);1(|

21

211

, ,1, , , , 1, , , ,

1, , , ,

1 , 1, , ,

( )( ) ( )

( 1 2 2 ) ( 2 2 )

( 2 2 2)( 2 2 1) ( 2 2 )( 2 2 1)

n u bt n u b n u b n u b n u b

x n u b n u b

n u b n u b

dpk d b p p b p p

dt

k n u b p n u b p

k n u b n u b p n u b n u b p

k

1 , 1, , ,

2 , 1, 1 , , 2 , 1, 1 , ,

( 1)

( 1)( ) ( ) ( 1)

n u b n u b

n u b n u b n u b n u b

u p up

k u d b p u d b p k b p bp

, |,

( , , ) ( , , ) |u b nu b

f u b n p f u b n n ,u b

Back to ODEIn the continuum limit (large n): Fokker-Planck equation

Corresponding Langevin equation

with )( txxx

(no prefactor)

Fast reaction noise is filtered out

Multiscale stochastic simulations(turbo-charged Gillespie algorithm)

• The computational analog of the projection procedure: stochastic partial equilibrium (Cao, Petzold, Gillespie, 2005):– Identify slow and fast variables – Fast reactions at quasi-equilibrium– distribution for fixed is assumed known– Compute propensities for slow reactions

( | )f f sP x x

( ) ( ( , ))s s s s fa E a x xx

,f sx x

sx

Easy for zero- and first-order reactions, more tricky for higher order reactions

Regulatory delay in genetic circuits

Single gene autoregulation: transcriptional delay

tk

1k

2k 1k2k

21 XXX k

XXX k 12

1202 DXD k

2012 XDD k

xkX

XPDPD tk 00

XPDPD tk 11

Fast

Slow

xkTtxKK

TtxKKdpkx xtx

)(1

)(12

21

221

0

P

DelayedAfter projection

[cf. Santillán & Mackey, 2001]

Genetic oscillations: Hopf bifurcation

Fixed point:

Complex eigenvalues

1,10 21 KKd

9/1crInstability

TktTkx

k tT

xd

xdHxGexdGkkx T

txx

)()()()( P

21 2

21 2

10 ( ) ( )

1x t

K K xk x k dH x H x

K K x

Instability

Transcriptional delay: a non-Markovian process

Markovian reactions [dimerization, degradation, binding]: • exponential “next reaction” time distribution

Non-Markovian channels [transcription, translation]: Gaussian time distribution

it

iT

2 20( ) exp ( ) /P

• which reaction to choose?

Stochastic simulations (modified Gillespie algorithm)

( ) expP t t a

aaP /)'( '

update update

Scheme of numerical simulation:delay

*t *t

time steps

Modified Direct Gillespie algorithm (Gillespie, 1977):1. Input values for initial state , set t=0 2. Compute propensities3. Generate random numbers4. Compute time step until next reaction 5. Check if there has been a delayed reaction scheduled in

a) if yes, then last steps 2,3,4 are ignored, time advances to , update in accordance with delayed reaction b) if not, go to the step 66. Find the channel of the next reaction: 7. Update time and

ixa

21, uu

aut /ln 1

ttt ,

dtt

ixttt

ix

1t2t 3t

if the reaction is delayed, postpone update until t t

1 1' ' 1

21 1 1 1

R R

a a u a a

Stochastic simulations

0,5.0,100,10 2121 KKkkd

Analytical resultsReactions: Deterministic model

No Hopf bifurcation!

Stochastic model (Master equations)

( , )sP n t probability to have n monomers at time t given the state s at time t-

Approximation: 0 1( , ; , ) ( , ) ( , ) ( , )s sP n t m t P n t P m t P m t

01 0 0 0 1

1 10

11 1 1 1 0

1 10

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

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

m

m

dP n tA E P n t B E P n t k mP n t m t k P n t

dt

dP n tA E P n t B E P n t k mP n t m t k P n t

dt

01 0 0 0 0 1 1

1 10

11 1 1 1 0 1 0

1 10

( , )( 1) ( , ) ( 1) ( , ) ( , )[ ( , ) ( , )] ( , )

( , )( 1) ( , ) ( 1) ( , ) ( , )[ ( , ) ( , )] ( , )

m

m

dP n tA E P n t B E P n t k mP n t P m t P m t k P n t

dt

dP n tA E P n t B E P n t k mP n t P m t P m t k P n t

dt

1 1/k k

1 10 1 1 0

0 0

,

,

k k

A B

D X D D D X

D D X X

( , ) ( 1, )EP n t P n t

(no dimerization)

Boolean model

Transition probability

)()(22

)( 2121 Ttstspppp

tp

1pp if )()( Ttsts

2pp if )()( Ttsts

Dp

4

41exp

2

322,1

ssp : at time t depends on the state at t-T:

21 pp positive feedback

21 pp negative feedback

For double-well quartic potential

Two-state gene:"on" ( 1)

"off" ( 1)

s

s

-1 1

)()()(

)()()(

12

21

TtnpTtnptW

TtnpTtnptW

)()()()()(

)()()()()(

tntWtntWtn

tntWtntWtn

Master equations)(tn the probability of having value s(t)= 1 at time t;

dttW )( s= 1 to 1dttW )( s= 1 to 1probability of transition from within (t,t+dt)

1 nn

Delayed master equation

)()()()()(

1221 TCppCppd

dC

Autocorrelation function

)()()()0()()( nnsssC

1)0();()( CCC Linear equation!

deTnC

ppnTCenTC

pp

pp

)')(('

0

12')(

21

21

))1((

)()()'(

Tpp

Tpppp

epppp

eppeppC

21

2121

21221

)(212

221)(

T allFor

T0For

Tn '0,...2,1

Autocorrelation function

T=1000, p1=0.1 p2=0.3

Stochastic oscillations!

Analytical resultsReactions: Deterministic model

No Hopf bifurcation!

Stochastic model (Master equations)

( , )sP n t probability to have n monomers at time t given the state s at time t-

Approximation: 0 1( , ; , ) ( , ) ( , ) ( , )s sP n t m t P n t P m t P m t

01 0 0 0 1

1 10

11 1 1 1 0

1 10

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

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

m

m

dP n tA E P n t B E P n t k mP n t m t k P n t

dt

dP n tA E P n t B E P n t k mP n t m t k P n t

dt

01 0 0 0 0 1 1

1 10

11 1 1 1 0 1 0

1 10

( , )( 1) ( , ) ( 1) ( , ) ( , )[ ( , ) ( , )] ( , )

( , )( 1) ( , ) ( 1) ( , ) ( , )[ ( , ) ( , )] ( , )

m

m

dP n tA E P n t B E P n t k mP n t P m t P m t k P n t

dt

dP n tA E P n t B E P n t k mP n t P m t P m t k P n t

dt

1 1/k k

1 10 1 1 0

0 0

,

,

k k

A B

D X D D D X

D D X X

( , ) ( 1, )EP n t P n t

(no dimerization)

Analytical resultsCorrelation function:

'

( ) ( ) ( ) ' ( , ; ', ) ', | ,0 ( )sn n n

K T n t n t T nn P n t n t T n n T n P n

Result:

Sta

nd

ard

dev

iati

on/m

ean

Time delay increases noise level

Effect of stochasticity and delay on regulation

Conclusions• Fast binding-unbinding processes can be eliminated both in

deterministic and stochastic modeling, however an accurate averaging procedure has to be used: leads to prefactors affecting transient times and noise distributions

• Multimerization increases time scales of genetic regulation

• Deterministic and stochastic description of regulatory delays developed, delays of transcription/translation of auto-repressor may lead to increased fluctuations levels and oscillations even when deterministic model shows no Hopf bifurcation

• Modified Gillespie algorithm is developed for simulating time-delayed reactions

L.S. Tsimring and A. Pikovsky, Phys. Rev. Lett., 87, 2506021 (2001).D.A. Bratsun, D. Volfson, L.S. Tsimring, and J. Hasty, PNAS, 102, 14593-12598 (2005).M. Bennett, D. Volfson, L. Tsimring, and J. Hasty, Biophys. J., 2007, in press.