# Polymerase Chain Reaction

## Polymerase Chain Reaction

• Developed by Kary Mullis in the 1980s
• used to synthesize millions of copies of a given DNA sequence

## PCR is based on

• Separate double-strand DNA using high temperatures, and
• Complementing each DNA single-strand using a polymerase that works at these high temperatures

## Taq DNA polymerase

Isolated from Thermus aquaticus

Can polymerize deoxynucleotide precursors (dNTPs) in a temperature range of 75-80°C

The polymerase extends a pre-existing pairing, initially made by primers that bind spontaneously to specific sites on the DNA

## A typical PCR reaction

A series of thermic cycles involving

• template DNA denaturation,
• primer annealing,
• and extension of the annealed primers by DNA polymerase

This three-step process is repeated 25-30 times

## Result

Exponential accumulation of a specific fragment

Ends defined by the 5’ ends of primers

## PCR reaction steps

### They are repeated 20-30 times

1. Denaturation of target DNA by heating to 90-95°C
2. Temperature is lowered to about 5°C below primers melting temperature
• assuring specificity of primer annealing
• assuring specificity of the product
3. Raising temperature to 70-73°C
• optimal temperature for primer extension

## Primers

Short DNA fragments with a defined sequence complementary to the target DNA that we want to detect and amplify

Each PCR assay requires the presence of template DNA, primers, nucleotides, and DNA polymerase

The primers in the reaction specify the exact DNA product to be amplified

# Melting temperature

## Melting temperature

The melting temperature ($$T_m$$) is the one in which half of the DNA molecules are matched and half are not matched

The $$T_m$$ shows the transition from double helical to random coil formation

It corresponds to the midpoint of the spectroscopic absorbance shift

## Measuring

The $$T_m$$ values are most simply measured by following the absorbance at 260 mM as a function of the temperature of the DNA solution and noting the midpoint of the hyperchromic rise

## Melting temperature depends on

• the DNA GC base content,
• the cation concentration of the buffer
• Salt concentration
• the DNA double strand length

## Salt concentration

The $$T_m$$ increases with higher concentrations of salt due to the stabilizing effects that cations have on DNA duplex formation

More cations bind to duplex DNA than to the single strands

Different cations may have different effects on $$T_m.$$

## Salt concentration

Most $$T_m$$ research is done using $$Na^+$$ as the primary cation; from a $$T_m$$ standpoint, sodium, and potassium are equivalent

Divalent cations (such as $$Mg^{++}$$) also increase $$T_m$$ but their effects are smaller than monovalent cations

## Oligonucleotide sequence

Sequences with a higher fraction of G-C base pairs have a higher $$T_m$$ than do AT-rich sequences

However, the $$T_m$$ of an oligo is not simply the sum of AT and GC base content

Base stacking interactions must also be taken into account such that the actual specific sequence must be known to accurately predict $$T_m$$

The effects of neighboring bases as contributed through base stacking are called “nearest neighbor effects”

# Formulas for $$T_m$$ calculation

## Formulas for $$T_m$$ Calculation

Different formulas have been developed, which can be classified into two groups:

• nucleotide composition
• position-dependent

Nucleotide composition-based formulas depend on the GC-content, the number of base pairs, and the salt concentration

Position-dependent methods depend on parameters such as enthalpy ($$ΔH^0$$), entropy ($$ΔS^0$$), and Gibbs free energy ($$ΔG^0$$)

## Wallace-Itakura Formula

It is one of the simplest ones, but it may not give the exact result $T_m = 4(G+C)+2(A+T)$ The formula was originally applied to the hybridization of probes in $$1 [mol/L]$$ of $$NaCl$$

This rule overestimates the $$T_m$$ of long duplexes

It gives reasonable results only in the range of 14-20 bp.

## Marmur and Doty Formula (1962)

The linear relation between the GC content and the $$T_m$$ was determined using absorbance shift analysis

For a solvent containing 0.2 Molar of Na+, the melting temperature is $T_m =69.3+0.41(\%GC)$ where $$T_m$$ is in degrees Celsius

The measurement of the $$T_m$$ is a way to determine the GC content of DNA

## Chester and Marshak Formula (1992)

Chester and Marshak added a term to account for DNA strand length (n in base pairs) to estimate primer $$T_m$$:

$T_m =69.3+0.41(\%GC)-\frac{650}{n}$

It is easy to see that if the DNA molecule is big (for example, if $$n>10^6$$), then this formula gives the same result as Marmur and Doty

## The Marmur-Schildkraut-Doty Equation (1964)

For ionic strength with a term for the $$Na^+$$ concentration

$T_m =81.5+16.6\log_{10}([Na^+])+0.41(\%GC)-\frac{b}{n}$

Values between 500 and 750 have been used for $$b$$ (a value that may increase with the ionic strength). Usually, the value $$b=500$$ is used.

## Wetmur Formula (1991)

\begin{aligned} T_m = & 81.5+16.6\log_{10}\left(\frac{[Na^+]}{1.0+0.7[Na^+]}\right) \\ & + 0.41(\%GC)-\frac{500}{n} \end{aligned}

This formula includes these variables with the salt concentration term modified to extend the range to 1 M $$Na^+$$, a concentration routinely employed to maximize hybridization rates on blots. (Wetmur, 1991)

## Position Dependent Calculation

These methods use thermodynamic parameters (entropy $$ΔS^0$$, enthalpy $$ΔH^0$$, and Gibbs free energy $$ΔG^0$$).

Experiments show that thermodynamic values for DNA melting do not depend only on base pair identity (A-T(U) or G-C)

Theoretical melting temperature is typically calculated assuming that the helix-coil transition is two-state

## Two state model

SantaLucia, et al. suggest that the two-state model gives a reasonable approximation of melting temperature for duplexes with non-two-state transitions

$\text{single-strand} + \text{single-strand} ⇆ \text{double-strand}$

## Thermodynamic Rule for $$T_m$$

For self-complementary oligonucleotide duplexes, $$T_m$$ is calculated from the predicted $$ΔH^0$$ and $$ΔS^0$$, and the total oligonucleotide concentration $$C_T$$, by using the equation $T_m=\frac{ΔH^0}{ΔS^0+R\ln (C_T)}$ where $$R$$ is the Boltzmann’s gas constant (1.987[cal/Kmol]) and temperature is measured in Kelvin degrees

(SantaLucia, 1998; Borer, Dengler, Tinoco, & Uhlenbeck, 1974)

## Energy

$$ΔG^0$$ is the free energy

Each $$ΔG^0$$ term has enthalpic, $$ΔH^0$$, and entropic, $$ΔS^0$$ components

The $$ΔG^0_{37}$$ can also be calculated from $$ΔH^0$$ and $$ΔS^0$$ parameters by using the equation: $ΔG^0_T=ΔH^0(\text{total})-TΔS^0(\text{total})$ (SantaLucia, 1998)

## Nearest-Neighbor Rule for energy

\begin{aligned} ΔG^0(\text{total})=∑_i n_iΔG^0(i)+ΔG^0(\text{init w/term G.C})\\ + ΔG^0(\text{init w/term A.T})+ΔG^0(\text{sym}) \end{aligned}

• $$ΔG^0(i)$$ are the free-energy changes for the 10 possible Watson-Crick Nearest-Neighbors
• $$ΔG^0(1)=ΔG^0_{37}(AA/TT),$$
• $$ΔG^0(2)=ΔG^0_{37}(AT/TA),$$ etc.
• $$n_i$$ is the number of occurrences of each Nearest-Neighbor
• $$ΔG^0(sym)$$ is $$+0.43\text{[k cal/mol]}$$ if the duplex is self-complementary and zero if it is not

## Formulas

$\underbrace{\text{single-strand DNA}}_A + \underbrace{\text{primer}}_B ⇆ \underbrace{\text{double-strand DNA}}_{AB}$

Equilibrium constant $$K$$ $K=\frac{[A][B]}{[AB]}=e^{-ΔG/RT}$ thus $-RT\log K = ΔG$

But we also have $ΔG=ΔH - TΔS$ so $TΔS-RT\log K = ΔH$ and therefore $T = \frac{ΔH}{ΔS-R\log K}$

Now $K=\frac{([A]_{ini}-[AB])([B]_{ini}-[AB])}{[AB]}$

Assuming that the initial concentration of primers is much larger than the initial DNA concentration, $$[A]_{ini}<<[B]_{ini},$$ and DNA will be the limiting factor. Thus, at the melting point we have $[AB]=[A]_{ini}/2$ thus \begin{aligned} K&=\frac{([A]_{ini}-[A]_{ini}/2)([B]_{ini}-[A]_{ini}/2)}{[A]_{ini}/2}\\ &=\frac{[A]_{ini}/2⋅ ([B]_{ini}-[A]_{ini}/2)}{[A]_{ini}/2}\\ &=[B]_{ini}-[A]_{ini}/2 \end{aligned}

So, at the melting point, we have $T_m = \frac{ΔH}{ΔS-R\log([B]_{ini}-[A]_{ini}/2)}$

Since the initial DNA concentration is small, we have $[B]_{ini}-[A]_{ini}/2 ≈ [B]_{ini}$ which gives us the final formula $T_m = \frac{ΔH}{ΔS-R\log([B]_{ini})}$