pcr modeling 2004.08.31 mec lim hee woong. denaturation temp. input melting curve of known dna conc....
TRANSCRIPT
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