System for determination of temperature cycling protocols for polymerase chain reactions; oligonucleotide annealing
In one aspect, methods are described herein for enhancing one or more nucleic acid interactions. For example, in some embodiments, methods of enhancing one or more steps of polymerase chain reaction (PCR) are described herein. In some embodiments, the optimal temperature cycling protocol for one or more PCR cycles can be determined according to methods described herein.
1. A system for enhancing annealing efficiency of first and second single strands of deoxyribonucleic acid (DNA) in a polymerase chain reaction (PCR) cycle at a predetermined time, the method comprising:
an amplification apparatus;
a memory device; and
a processor coupled to the memory device, the processor configured to:
(a) receive information comprising sequences of single strand DNA and primer, concentrations of the single strand DNA and primer, length of a predetermined time period for annealing the single strand DNA and primer and a set of annealing reaction temperatures;
(b) determine the forward reaction rate constant, k f , of annealing the single strand DNA and primer at annealing reaction temperatures in the set;
(c) determine the reverse reaction rate reaction rate constant, k r , of annealing the single strand DNA and primer at reaction temperatures in the set; and
(d) select one or more of the annealing reaction temperatures sufficient to produce a primer hybridized DNA strand concentration in excess of a predetermined threshold within the predetermined time period for annealing; and
at least one output interface for communicating the one or more annealing reaction temperatures to the amplification apparatus, wherein the amplification apparatus conducts the annealing to produce the primer hybridized DNA strand concentration in excess of the predetermined threshold within the predetermined time period,
wherein the hybridized DNA strand concentration, [SP], is determined according to:
d [ SP ]/ dt=k f [ S ][ P ]− k r [ SP ];
wherein [P] is concentration of the primer and [S] is concentration of the single strand DNA; and
wherein k f and k r are calculated according to Gibbs free energy for the annealing reaction and relaxation time for the annealing reaction, and wherein relationship between the Gibbs free energy, k f and k r is expressed as:
k
f
k
r
=
[
C
o
]
exp
(
-
Δ
G
RT
)
wherein [C 0 ] is the standard concentration, R is the gas constant and T is the annealing reaction temperature and the relaxation time τ is expressed as:
τ=1/( k f [ S 2 eq +S 1 eq ]+ k r )
wherein S 1 eq and S 2 eq are equilibrium concentrations of the first and second DNA single strands.
2. The system of claim 1 further comprising at least one input interface for entry of the information for the determination of k f and k r .
3. The system of claim 1 further comprising at least one output interface for communicating the selected annealing reaction temperatures to a user of the system or to apparatus for conducting the annealing reaction.
4. The system of claim 1 , wherein the relaxation time is expressed as τ=−1/λ max
wherein λ max denotes the largest eigenvalue of an annealing reaction matrix wherein an annealing reaction matrix for a two-base sequence is of the form:
A
=
[
-
(
k
-
1
+
k
-
1
)
k
1
k
1
0
k
-
1
-
(
k
-
2
+
k
1
)
0
σ
[
D
0
,
2
eq
]
k
1
k
-
1
0
-
(
k
2
+
k
1
)
σ
[
D
0
,
2
eq
]
k
1
0
k
-
2
k
-
2
-
2
σ
[
D
0
,
2
eq
]
k
1
]
where:
[D 0,2 eq ] denotes the equilibrium concentration of fully dissociated DNA,
k 1 =k 2 =k 1 =k 2 which equal the forward rate constant for formation of a single base pair,
k −i is the reverse rate constant for the formation of the single base pair i bases from the DNA molecule's left side wherein k −i =k 1 /s i where s i is the equilibrium constant for formation of the base pair (respectively for k −1 and the base pair i bases from the DNA molecule's right side),
and σ is a nucleation parameter that measures resistance to formation of the first base pair.
5. A system for enhancing annealing efficiency of first and second single strands of deoxyribonucleic acid (DNA) in a polymerase chain reaction (PCR) cycle at a predetermined time, the method comprising:
an amplification apparatus;
a memory device; and
a processor coupled to the memory device, the processor configured to:
(a) receive information comprising sequences of single strand DNA and primer, concentrations of the single strand DNA and primer, length of a predetermined time period for annealing the single strand DNA and primer and a set of annealing reaction temperatures;
(b) determine the forward reaction rate constant, k f , of annealing the single strand DNA and primer at annealing reaction temperatures in the set;
(c) determine the reverse reaction rate reaction rate constant, k r , of annealing the single strand DNA and primer at reaction temperatures in the set; and
(d) select one or more of the annealing reaction temperatures sufficient to produce a primer hybridized DNA strand concentration in excess of a predetermined threshold within the predetermined time period for annealing; and
at least one output interface for communicating the one or more annealing reaction temperatures to the amplification apparatus, wherein the amplification apparatus conducts the annealing to produce the primer hybridized DNA strand concentration in excess of the predetermined threshold within the predetermined time period,
wherein the hybridized DNA strand concentration, [SP], is determined according to:
d [ SP ]/ dt=k f [ S ][ P ]− k r [ SP ]
wherein [P] is concentration of the primer and [S] is concentration of the single strand DNA; and
wherein k f and k r are calculated according to Gibbs free energy for the annealing reaction and relaxation time for the annealing reaction, and wherein relationship between the Gibbs free energy, k f and k r is expressed as:
k
f
k
r
=
[
C
o
]
exp
(
-
Δ
G
R
T
)
wherein [C 0 ] is the standard concentration, R is the gas constant and T is the annealing reaction temperature and the relaxation time τ is expressed as:
τ=1/( k f [ P eq +S eq ]+ k r )
wherein P eq and S eq are equilibrium concentrations of the primer and single strand DNA.
6. The system of claim 5 , wherein k f and k r are calculated according to Gibbs free energy for the annealing reaction and relaxation time for the annealing reaction, and wherein relationship between the Gibbs free energy, k f and k r is expressed as:
k
f
k
r
=
[
C
o
]
exp
(
-
Δ
G
RT
)
wherein [C 0 ] is the standard concentration, R is the gas constant and T is the annealing reaction temperature and the relaxation time τ is expressed as:
τ=1/( k f [ P eq +S eq ]+ k r )
wherein P eq and S eq are equilibrium concentrations of the primer and single strand DNA.
7. The system of claim 6 , wherein the relaxation time is expressed as τ=−1/λ max wherein λ max denotes the largest eigenvalue of an annealing reaction matrix wherein an annealing reaction matrix for a two-base sequence is of the form:
A
=
[
-
(
k
-
1
+
k
-
1
)
k
1
k
1
0
k
-
1
-
(
k
-
2
+
k
1
)
0
σ
[
D
0
,
2
eq
]
k
1
k
-
1
0
-
(
k
2
+
k
1
)
σ
[
D
0
,
2
eq
]
k
1
0
k
-
2
k
-
2
-
2
σ
[
D
0
,
2
eq
]
k
1
]
where:
[D 0,2 eq ] denotes the equilibrium concentration of fully dissociated DNA,
k 1 =k 2 =k 1 =k 2 which equal the forward rate constant for formation of a single base pair,
k −i is the reverse rate constant for the formation of the single base pair i bases from the DNA molecule's left side wherein k −i =k 1 /s i where s i is the equilibrium constant for formation of the base pair (respectively for k −1 and the base pair i bases from the DNA molecule's right side),
and σ is a nucleation parameter that measures resistance to formation of the first base pair.