PCR Modeling

2004.08.31MEC

Lim Hee Woong

Denaturation

Temp.input

Melting curve of

known DNA conc.

Keq

output

dsDNA conc.input

Released ssDNA conc.output

denaturationefficiency

• Melting Profile– Sigmoid function assumption

– Parameters• Tm: Melting temperature : Transition width: begin & end temperature• Cinit: initial dsDNA concentration for melting profile• Measured and fitted by experiment with UV spectrophotomete

r or real-time PCR machine

1

ratio of denatured dsDNA

1mT T

T

e

• Equilibrium Constant Kd from Tm Profile

2 2

2

/ 2

4

1

ss ssd

ds init ss

init

C CK T

C C C

T C

T

2

1

ss init

ds init

C T C

C C T

2 2

0

2

0

2

0 0

2

0

0 0

/ 2

16

4

8 16

8

16/ 2

8

ss ssd

ds ds ss

d d d dsss

ds d d d dsds

d d d dsssd

ds ds

C CK T

C C C

K T K T K T CC

C K T K T K T CC

K T K T K T CCE

C C

• Denaturation– Cds0: initial dsDNA strand in denature step– Kd(T): equlibrium constant from melting profile

2

0

0

0

0

0

0

0

16

8

16

8

16

8

21

16

d d d dsd

ds

d d d ds

ds

d d ds d

ds

d

d ds d

K T K T K T CE

C

K T K T K T C

C

K T K T C K T

C

K T

K T C K T

Annealing

• 경쟁적인 annealing 반응 , primer vs. template

1

2'

k

k

Primer Template hdDNA

Template Template dsDNA

• 다른 step (denaturation, extension) 과는 달리 각 template 에 대해 forward 와 backward를 구분

– 대칭성에 의해 forward 와 backward strand 의 농도변화는 같다고 가정– 위의 계산식에 사용되는 Css, 와 Chd 는 forward 혹은 backward 의 한쪽 방향 strand 에 대한

농도– 앞의 denaturation step 에서 계산된 ssDNA 의 농도의 절반 만큼의 농도를 할당하여 계산

수행하고 Annealing 이 끝난 후 위 계산 결과에 2 를 곱하여 backward/forward 의 구분을 없앰• Rate constant k 의 값을 몰라도 그 비율만으로 최종 product 의 비율을 계산 가능

– Wetmur (Annu. Rev. Biophys. Bioeng. 1976)• Calculation

– Numerical method, Runge-Kutta formulae– Matlab function “ode45”

,1 ,0 , ,0 , ,

2,2 ,0 , ,

,0 ,00, 0

hd tpr hd t ss hd t ds t

ds tss hd t ds t

hd ds

dCk C C C C C

dtdC

k C C CdtC C

strandlonger oflength : N

strandshorter oflength :LN

Lk

Wetmur (Annu. Rev. Biophys. Bioeng. 1976)

Annealing 2-Temperature Ramping-

Temperature

Time

Template Tm

hdDNA Tm

Pre-annealingof template

Competitive annealingto form dsDNA and hdDNA

• Tm,hdDNA<Tm,dsDNA– Pre-annealing takes place before competitive annealing

'2' kTemplate Template dsDNA

1

2'

k

k

Primer Template hdDNA

Template Template dsDNA

Pre-annealing

Competitive annealing

annealing

,1 ,0 , ,0 , ,

2,2 ,0 , ,

,0 0

hd tpr hd t ss hd t ds t

ds tss hd t ds t

hd

dCk C C C C C

dtdC

k C C CdtC

strandlonger oflength : N

strandshorter oflength :LN

Lk

Wetmur (Annu. Rev. Biophys. Bioeng. 1976) 2,

2 ,0 ,

,0 0,

ds tss ds t

ds

dCk C C

dtC

Extension

• rt: t 초 후의 reaction rate• ke: extension rate for one polymerase• knu: nucleotide incorporation rate of one polymerase• Ea,t: t 초 후에 실제 polymerization 에 참가하는 enzyme 의 농도

trhdDNA dsDNA

,t e a t

nue

r k E

kk

length

From Hsu et al.BB 1997

• Active enzyme– Cenz: thermal deactivate 되지 않고 남아있는 polymerase 농도– Cenz 를 hetero duplex 부분과 template duplex 부분의 비율로 나누어 실제 polymerization 에

참가하는 enzyme 의 비율을 구한다 .– 단순히 duplex 의 농도만 고려하는 것이 아니라 duplex region 의 길이까지도 고려함– Kainz (BBA 2000) paper 참조

• Calculation– Numerical method, Runge-Kutta formulae– Matlab function “ode45”

,,,

, ,

0 ,

0 ,

0 , , ,0 ,0

p hd tds te a t e enz

p hd t ds t

p ds t

e enz

p p ds t

hd t ds t hd ds

l CdCk E k C

dt l C lC

l C Ck C

l C l l C

C C C C C

,,

, ,

p hd ta t enz

p hd t ds t

l CE C

l C lC

실제 polymerization 에 참가하는 enzyme 의

농도

Enzyme Deactivation

• Hsu et al. (BB, 1997) 논문의 deactivation 식에 temperature ramping 을 추가하여 확장

• at: remaining enzyme ratio after t second– considering temperature ramping

• T1, T2, t: t∆ ∆ 초 동안 T1 에서 T2 로 온도가 변함• Calculation

– Numerical method, Runge-Kutta formulae– Matlab function “ode45”

112

0

'00

0

'expln

law Arrhenius

ondeactivatiorder first

Ttt

TTtT

dteKaeKdt

ad

eKK

aKdt

da

ttRT

E

dttRT

E

dt

tRT

E

dd

tdt

dd

d

K/mol cal 987.1K J/mol 31441.8

J/mol1032.7

10677.45

1000

R

E

K

d

d

From Hsu et al.BB 1997

Product Flow

By- Productand

Surplus

Main Streamand

Expected Product

Reagentand

Reaction Step

Denature

Annealing

Extension

dsDNAHeteroDuplex

dsDNAHeteroDuplex

Polymerase

dsDNA ssDNA Primer

Primer

dsDNA

Simulation

9e-012 4.93815e-008

9e-013 4.28836e-008

9e-014 3.59484e-008

PCR Plateau?

• When varying amounts of a single target are amplified, a constant maximum level of product is obtained.

• Coamplification of different concentrations of different targets results in retention of the initial proportions.

• Morrison et al. BBA, 1994

Figuresfrom TAKARA

Factors of Plateau?

• Reduction in the denaturation efficiency• Utilization of substrates (dNTPs or primers)• Reannealing of specific product at concentrations above

10-8 M• Thermal inactivation or limited concentration of DNA poly

merase• Exonuclease activity of Taq polymerase• Inhibition of enzyme activity by increasing pyrophosphate

In Plateau• In plateau

– The template concentration reaches constant level (about 10-8 order), even if the order of initial concentration varies.

• What occurs at each step in plateau?– Denaturation

• Constant denaturation efficiency and ssDNA concentration• ≈ 1, almost perfect denaturation• Not only in plateau

– Annealing• Constant annealing efficiency and hdDNA concentration• ≈ 0 or >> 0 ?

– Extension• Constant extension efficiency• ≈ 0 or >> 0 ?

• Question…– Annealing efficiency and the amount of hdDNA in plateau– Extension efficiency in plateau– What is the major factor for plateau

Other Insignificant Factors

• Mis-annealing of primers• Reaction condition change

– pH change?– MgCl2 concentration?– DNA contaminants (non-specific products, pri

mer-dimer)

Parameters to Fit

• Hybridization rate constant• Extension rate constant• Pre-annealing (?) region

Extension2

• Michaelis-Menten Equation + BB paper

1 3

2

4

5

1

2

1 4

k k

k

k

k

hdDNA Enzyme Complex dsDNA Enzyme

dsDNA Enzyme Complex

k k

,1 , , 2 ,

,1 , , 2 , 3 ,

, , ,0

, , , , ,01 , , 3 , 2 ,

,3 ,

hd thd t enz t c t

c thd t enz t c t c t

enz t c t enz

enz t hd t c t ds t hdhd t enz t c t c t

ds tc t

dCk C C k C

dtdC

k C C k C k C C C CdtdC C C C C

k C C k C k Cdt

dCk C

dt

,1 , , 2 ,0 ,

,1 , , 3 2 ,0 ,

hd thd t enz t enz enz t

enz thd t enz t enz enz t

dCk C C k C C

dtdC

k C C k k C Cdt