8/10/2019 Poisson Interval

1/8

Poisson Interval Distributionand

Poisson Distributions

Random Walks discussionby Joshua Wayment and

Ryan White

2/15/05

8/10/2019 Poisson Interval

2/8

Poisson Interval Distribution

0 t

n

2t

n

3t

n

t t dt+

Single event process:

Probability of an event observed between t and t+dt

Divide time interval into large number of increments, n,

of length t/n.

P(t;!)= !t//n assume n>>1, gives greaterprobability of only a single event

Probability an event does not occur in first increment

.= 1-!t/n

Probability an event does not occur in n successive increments

= (1-!t/n)n

Probability an event does occur in the next time increment dt

= !dt

8/10/2019 Poisson Interval

3/8

8/10/2019 Poisson Interval

4/8

Poisson Distribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10

k

P(k;)

Events in a given amount of time:

Probability of an event does not occur

10

! ="

! ! "

#t

t te dt e$

$ $

Probability an event occurs in interval t

0 0

0 0!

" " !" " !

# = !t

t t t t e e dt t e$ $

$ $ $( )

Probability of two events

( ) ( )

0 0 0

2

0 0

2

!" " ! " " !

! " = !t

t t t t

e t t e dt

t

e$ $

$$ $ $( )

Probability of k events: ( )P kk

e

k

;!

= !

8/10/2019 Poisson Interval

5/8

!

)(),(

k

tetkp

kt!

=!

!"#

#

#

Ct = integration timek = number of counts

!= 1/"= average capture rate

ttp !=! "" )),0(ln(

slope=!

!= 0.98 s-1

-!= -8.5

What it should be

What it is

Particle Capture Statistics

integration time (s)

ln(p

(0))

capture time (s )

frequency

8/10/2019 Poisson Interval

6/8

5-(and-6)-carboxytetramethylrhodamine

succinimidyl ester (TAMRA)

Common IgG

For our purposes we use

monoclonal anti-biotin from

mice, conjugated with TAMRA

Single molecule studies of antibody/ antigen binding.

Using TAMRA conjugated IgG we can investigate single

molecule kinetics for association and disassociation.

Meaning that instead of making an ensemble measurement

we are examining kinetics on a single molecule basis

Anti-biotin IGG binding to immobilized biotin molecules

8/10/2019 Poisson Interval

7/8

0.0 0.5 1.0 1.5 2.0

0

10

20

30

40

50

Data: Data1_BModel: ExpDec2

Chi 2 = 2.83486R^2 = 0.94187y0 0 0A1 145.56279 29.02043t1 0.04823 0.0062A2 21.03456 2.65471t2 0.33086 0.0295

N

umberofObservations

Time to bind (s)

.25 g/ml

Binding kinetics (biphasic)

0

5

10

15

20

25

30

35

40

45

50

0 0.5 1 1.5 2 2.5 3 3.5

[Ab] (nM)

BindingRate(s)

kbind~ 1.3 x 1010M-1s -1

kbind~ 1.4 x 109

M-1

s-1

Binding kinetics modeled using two

exponential system:

y=A1exp(-t1x)+ A2exp(-t2x)

What does it mean to have biphasic kinetics?

Anti-biotin has two possible routes of finding biotin

1) Diffusion through solution

2) Diffusion on surface

8/10/2019 Poisson Interval

8/8

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5

[Ab] (nM)

UnbindingRate(s)

Unbinding kinetics

0.0 0.2 0.4 0.6 0.8 1.0

0

10

20

30

40

Data: Data1_DModel: ExpDec1

Chi 2 = 2.70126R^2 = 0.9393y0 0 0A1 173.5066 16.45395t1 0.04749 0.00232

NumberofObservation

s

Time to Unbind (s)

.25 g/ml

Unbinding kinetics modeled using single

exponential

y = A1exp(-t1x)

Unbinding kinetics

1) Simple one step process

2) Independent of concentration