Stochastic models of chemical kinetics

5. Poisson process

... a single strand of RNA is synthesized using a double stranded DNA molecule as a template.

The two strands of the DNA molecule are separated from one another, exposing the nitrogenous bases. Only one strand is actively used as a template in the transcription process, this is known as the sense strand, or template strand.

The RNA sequence that is made is a direct copy of the nitrogenous bases in the sense strand.

If an Guanine (G) base is part of the sequence on the sense DNA strand, then the RNA molecule has a Cytosine (C) base added to its sequence at that point. In the RNA molecule uracil (U) substitutes for Thymine (T)

... is catalyzed by an enzyme called RNA polymerase, which needs as substrates double stranded DNA, and the ribonucleotides ATP, UTP, CTP and GTP.

One at a time, this enzyme adds ribonucleotides to a growing RNA strand by joining incoming ribonucleotide triphosphates to the ribose sugar molecule of the last nucleotide of the growing RNA strand.

Two of the phosphate groups are removed from the triphosphate and a covalent bond is formed between the remaining phosphate and the third carbon atom of the ribose sugar at the end of the RNA strand.

Initiation of the transcription process begins with binding of the RNA polymerase enzyme to the DNA molecule at a region known as the promoter site. This site is right in front of a gene where transcription will begin.

The RNA polymerase enzyme does not copy the promotor into the RNA, but begins the synthesis of the RNA at a specific nucleotide sequence called the start signal or initiation site which is often the bases GTA on the DNA (which then become the bases CAU on the RNA molecule).

Model of transcription: Poisson process

1 2

1. (Homogeneity) Rate of transcript generation is constant:

the exp

Transcripts generated at random t

ected number of transcripts gener

imes , ,...in [0

ated in a subin

, ].

Assumption

terval

of len

s

th

:

g

X X t

2. (Independence) Numbers of transcripts generated in disjoint

time intervals are independent random var

is

i e

s.

.

abl

u u

time0 1X 2X 3X 4X 5X

Number of events in time t

Use this to get the distribution for

Number of transcripts generated in time , ( ) or

Homogeneity: [ ] .

1. Separate [0, ] into intervals, each of length / .

2. Since intervals are

:

small,

t

t

t

t N t N

E N t

t n t

N

n

each of them contains either 0 or 1 events

(transcripts generated). Each interval is characterized by a Bernoulli

distribution, with the probability of success, .t

pn

Number of events in time t

3. Number successes in trials:

! !( ) (1 ) 1

! ! !

( 1) (

!

!

1)lim ( ) lim 1 1

lim 1 .! !

n kk n k

t

n k

t kn n

k k

k

k

nt

n

t

n

n n tP N k p p

k n k n k n n

n n n k t tP N k

n n n

t tte

k

t

k

k n k

( )

!

k

tt

tP N k e

k

Distribution of times between successive events

0

Probability that nothing happens in [0

events happened by time .

What is th

, ]:

( )( 0, )

0

Probability that the first event happens i

e probabili

n [0, ]

ty of the nex

!

t event in [ , ]?

:

t t

N s

s s

t

t

tP k t e e

t

P

( 0, ) 1( , )1 0 tP kk t et

Distribution of times between successive events

( )

( )

rate of transition

from a state with events

i

Probability of an event in [ ,

nto a state with 1 event

]:

( ) 1 (1 )

s

t dt t

t t dt t

t t dt

P t T t dt e e

e e

dt

N

N

dt e

Simulating the Poisson process (1)

(

S

n

ta

ex

rt w

t ev

ith molecules,

ent

then

Need to generate r.v. fro

find time for the 1 tran

m ( ) 1 ( )

)

on:

1

siti

t

tP t

N

N N t

T P T t e F t

h

e

Simulating the Poisson process (2)

1

1. generate : P

2. ( ) ( ) are equivalent events

3.Thus: ( ) ( )

4.Thus, ( ) is distributed according

( ) P ( )

( ) :

to ( )

log 1

log 1, .

1

1

inv

inv

inv

in

i

i

v

nv

t

i

U U u u

U F t F U t

P U F t P F U t

F t F U t

F U

F U

u e

F t

ut

uT T N N

0

1

u

t( )invF u

( )F t

More complex model:birth-death process

• Transcripts are generated according to the Poisson process, with intensity

• Transcripts are degraded with the rate constant,

• We are interested in simulating the dynamics of this system

• Start with a number of molecules and advance the state of the system

• Need to determine: the time of the next event and the type of the event

• In this system: two types of reactions: generation and degradation

M

Survival of a state with n molecules

( , ) : probability that there are molecules at time

Synthesis and degradation reactions are independent.

Probability that happens

in [ , ] is

1 ( ) : pr

either one of the

obability o

m

f

P n t n t

t t dt dt ndt

kn dt

( , ) ( , ) (1 ( )

no events in [ , ]

)

P n t

t t dt

dt P n t kn dt

Lifetime of a state with n molecules

( , )

( ) ( , )( , )

( ,0) 1

1 ( , ) : Probability that reaction event happened be

Thus, the next reaction (transcription or tr

fo

a

r

n

(

script degrad

e

, ) ( , ) (1 (

a

.

) )

n t

dP n tn P n t

P n t edtP

P n t dt P n

n

P n t

t kn t

t

d

tion)

occurs at a random time : { } 1 n tT P T t e

Choosing the event (1)

: the system survived until time with

molecules and the next event is transcript

(d

(transcription, )

(transcription, ) ( ,

egradation,

ion

: the system survived until time with

mo

)

le

) 1

tp t

p t dt P n t n

n

t

n

p

n

t

cules and the next event is transcript degra

(transcription,

dation

) ( , ) 1p t ndt P n t n n

Choosing the event (2)

(transcription degradation in ( , ))

( , ) ( , )

( , )(transcription)

( , ) ( , )

( , )(degradation)

or

disjoint events

( , ) ( , )

p t t dt

dt P n t ndt P n t

dt P n tp

dt P n t ndt P n t n

ndt P n t np

dt P n t ndt P n t n

Simulation algorithm (1)

( )

molecules at time

1. Choose T according to

( ) 1

2. Choose event according to Ber

(transcription) 1 (degradation)

noulli dis

(de

tributi

gradation

on

) 1 (transcri ti

1

p on)

n

n t

P T e t t

p p

T

p p n nn

1n

n nn

Simulated trajectories

Probability Distribution at different times

Simulation algorithm (2)

1. Can be generalized for an arbitrary number of reactions and species

2. Use it when master equation is difficult to solve

3. Many extensions for computational efficiency and spatial effects

4. Whenever you have a code, test it using analytically solvable problems

Gillespie, D.T. Exact stochastic simulation of coupled chemical reactions.

JOURNAL OF PHYSICAL CHEMISTRY, 1977    Vol: 81(25)    pp: 2340-2361

(read! This citation classic has a very good description of the algorithm)

Gene expression model:Master equation

,00

00 1

1 1

( ) is the distribution function

for the number of particles

accumulated by time t; 0,1,2,

( ) 1; ( 0

1,2, , ;

Infinitely many diffe

)

( ) ( 1)

r

:

n n

n n nn

nn n n

p t p

n

p t

n

p t

dpp p

dtdp

p n p n pdt

ential equations.

M

First moment

HW: chec

0

10

k

10

( ) ( ), (0) 0

( 1)

(1 ) (1 )

nn

nn n n n

n n

t t

ss

n t n np t n

d n dpn n p p np n p

dt dt

n

d nn n e n e

dt

Distribution function

2

0 1 2

0

0 1

, where

( ) , ( ) , ( ) ,2

Check:

(

( )

)!

!

n n n

n

n

n

n

n n n

n

d nn

dt

np t e p t n e p t e etc

d ndp de

np t e

n

edt dt dt

n e e n e

p p

Experimental example

Visualizing mRNA molecules in cells

(A) Schematic depicting the mRNA detection method. Multiple fluorescent probes bind to each mRNA molecule, yielding a bright, localized signal.(B) Merged image of a three-dimensional stack of images from a CHO cell. Each spot corresponds to a single mRNA molecule.(C) Identification of the spots in the three-dimensional image stack in (C). Each particle found by the image-analysis algorithm is colored differently, showing that the algorithm is accurate and that individual molecules are uniquely identified. The scale bars are 5 μm long.

mRNA copy numbers in cells

Histograms showing the distribution of mRNA molecules per cell for three levels of gene activity.