Physical origin of DNA unzipping

Abstract

In DNA transcription, the base pairs are unzipped in response to the enzymatic forces, separating apart two intertwined nucleotide strands. Consequently, the double-stranded DNA (dsDNA), in which two nucleotide strands wind about each other, transits structurally to the single-stranded DNA (ssDNA) in which two nucleotide strands are completely unwound and separated. The large interstrand separation is intimately related to the softening nucleotide strands. This conceptual framework is reinforced with the flow of the bending modulus toward zero under recursion relations derived from the momentum shell renormalization group. Interestingly, the stretch modulus remains the same under recursion relations. The renormalization of the bending modulus to zero has a profound implication that ssDNA has the shorter bending persistence length than does dsDNA in accordance with experiments.

Appendix: Derivation of recursion relations by momentum shell renormalization group

We present the detail of calculation leading to the recursion relations, (17), governing how the stretch modulus A, bending modulus B, helix wavevector k ₀, and the force components F _x ,F _y change as approaching to the dsDNA-to-ssDNA transition. The elastic properties, like stretch modulus A and bending modulus B, are insensitive to the short distance structure, i.e., large wavevector q. The dsDNA-to-ssDNA transition can be equally described by the Hamiltonian, which involves only the small wavevector q but with the parameters in the Hamiltonian being renormalized. The large wavevector corresponds to the high momentum. Integrating out the high momentum part of the field r ^>(q) appearing in the Hamiltonian H ^>, (15), leads to the new Hamiltonian H _Λ/b [r ^<(q)], involving only the low momentum part of the field r ^<(q) with small wavevector 0<q<Λ/b

$$ \exp\left( -\frac{ H_{\frac{\Lambda}{b} }[\textbf{r}^{<}(q)] }{k_{B}T} \right) = \exp\left( -\frac{ H^{<}[\textbf{r}^{<}(q)] }{k_{B}T} \right) \int D\textbf{r}^{>}(q) \exp\left( -\frac{ H^{>}[\textbf{r}^{<}(q),\textbf{r}^{>}(q)] }{k_{B}T} \right). $$

(21)

The new Hamiltonian therefore takes the form

$$\begin{array}{@{}rcl@{}} H_{\frac{\Lambda}{b} }[\textbf{r}^{<}(q)] & = & \frac{1}{8} {\int}_{\!\!\!\!0}^{\frac{\Lambda}{b} } \frac{dq}{2\pi}\left[ A\left( q^{2} + {k_{0}^{2}} \right) + B\left( q^{4} + {k_{0}^{4}} \right) \right] x^{<}(q) x^{<}(-q) \\ & & + \frac{1}{8} {\int}_{\!\!\!\!0}^{\frac{\Lambda}{b} } \frac{dq}{2\pi}\left[ A\left( q^{2} + {k_{0}^{2}} \right) + B\left( q^{4} + {k_{0}^{4}} \right) \right] y^{<}(q) y^{<}(-q) \\ & & - \frac{1}{2} {\int}_{\!\!\!\!0}^{\frac{\Lambda}{b} } \frac{dq}{2\pi} \left( F_{x}x^{<}(q) + F_{y}y^{<}(q) \right) + D \end{array} $$

(22)

where the constant is

$$ D = -{\int}_{\!\!\!\!0}^{\frac{\Lambda}{b} } \frac{dq}{2\pi} \frac{ {F_{x}^{2}} + {F_{y}^{2}} } { A\left( q^{2} + {k_{0}^{2}} \right) + B\left( q^{4} + {k_{0}^{4}} \right) } + k_{B}T{\int}_{\!\!\!\!0}^{ \frac{\Lambda}{b} } \frac{dq}{2\pi} \ln\frac{ A\left( q^{2} + {k_{0}^{2}} \right) + B\left( q^{4} + {k_{0}^{4}} \right) }{4\pi k_{B}T}. $$

(23)

Note that the second term in D is −k _B T lnC, recalling the constant C from (9). To restore the momentum cutoff Λ/b back to Λ, the momentum is rescaled by q=q ^′/b and the field is rescaled by r ^<(q)=ζ r ^′(q ^′), corresponding to, in real space, the arc length rescaled by s=b s ^′ and the field rescaled by r ^<(s)=(1/ζ)r ^′(s ^′). This momentum and field rescaling results in a new Hamiltonian $H_{\Lambda }^{\prime }[\textbf {r}^{\prime }(q^{\prime })]$, but now with the same momentum cutoff Λ as does the original Hamiltonian H,

$$\begin{array}{@{}rcl@{}} H_{\Lambda}^{\prime}[\textbf{r}^{\prime}(q^{\prime})] & = & \frac{\zeta^{2}}{8b^{3}} {\int}_{\!\!\!\!0}^{\Lambda} \frac{dq^{\prime}}{2\pi} \left[ A\left( q^{\prime 2} + b^{2}{k_{0}^{2}} \right) + \frac{B}{b^{2}} \left( q^{\prime 4} + b^{4}{k_{0}^{4}} \right) \right] x^{\prime}(q^{\prime}) x^{\prime}(-q^{\prime}) \\ & & + \frac{\zeta^{2}}{8b^{3}} {\int}_{\!\!\!\!0}^{\Lambda} \frac{dq^{\prime}}{2\pi} \left[ A\left( q^{\prime 2} + b^{2}{k_{0}^{2}} \right) + \frac{B}{b^{2}} \left( q^{\prime 4} + b^{4}{k_{0}^{4}} \right) \right] y^{\prime}(q^{\prime}) y^{\prime}(-q^{\prime}) \\ & & - \frac{\zeta}{2b} {\int}_{\!\!\!\!0}^{\Lambda} \frac{dq^{\prime}}{2\pi}\left[ F_{x}^{\prime}x^{\prime}(q^{\prime}) + F_{y}^{\prime}y^{\prime}(q^{\prime}) \right]. \end{array} $$

(24)

This new Hamiltonian $H_{\Lambda }^{\prime }[\textbf {r}^{\prime }(q^{\prime })]$, (24), has the identical form of the original Hamiltonian H[r(q)], (6), suggesting that the stretch modulus A, bending modulus B, helix wavevector k ₀, and the force components F _x ,F _y must be renormalized according to

$$ \begin{array}{ccccc} A^{\prime} = \frac{\zeta^{2}}{2b^{3}}A, \hspace{5mm} & B^{\prime} = \frac{\zeta^{2}}{2b^{5}}B, \hspace{5mm} & k_{0}^{\prime} = bk_{0}, \hspace{5mm} & F_{x}^{\prime} = \frac{\zeta}{2}F_{x}, \hspace{5mm} & F_{y}^{\prime} = \frac{\zeta}{2}F_{y}. \end{array} $$

(25)

The rescaling parameter ζ for the field is chosen to satisfy the scaling law of correlation. The inverse of the correlation $G_{ii}^{-1}(s, s^{\prime })$ is the second functional derivative evaluated at the mean-field solution <r _i (s)> [13]

$$ G_{ii}^{-1}(s, s^{\prime}) = \frac{1}{k_{B}T} \left. \frac{ \delta^{2}H }{ \delta r_{i}(s) \delta r_{i}(s^{\prime}) } \right|_{r_{i}(s) = \left< r_{i}(s) \right> }. $$

(26)

Using the relative-coordinate part of Hamiltonian (5) gives

$$ \frac{ \delta^{2}H }{ \delta r_{i}(s) \delta r_{i}(s^{\prime}) } = -\frac{A}{2} \frac{d^{2}}{ds^{2}} \delta(s - s^{\prime}) + \frac{B}{2} \frac{d^{4}}{ds^{4}} \delta(s - s^{\prime}) + \frac{ {k_{0}^{2}} }{2} (A + B{k_{0}^{2}}) \delta(s - s^{\prime}). $$

(27)

With the Fourier transform of the delta function, $\delta (s - s^{\prime }) = \int (dq/2\pi ) \exp (iq(s - s^{\prime }) )$, substitute (27) in (26) gives the inverse of the correlation

$$ G_{ii}^{-1}(s, s^{\prime}) = \int\frac{dq}{2\pi} \frac{1}{k_{B}T} \left( \frac{A}{2} (q^{2} + {k_{0}^{2}}) + \frac{B}{2} (q^{4} + {k_{0}^{4}}) \right) \exp(iq(s - s^{\prime}) ) $$

(28)

which is the Fourier transform of $G_{ii}^{-1}(s, s^{\prime })$. Apart from exp(i q(s−s ^′)), the integrand is the inverse of the correlation in momentum space $G_{ii}^{-1}(q)$, resulting in the correlation

$$ G_{ii}(q) = \frac{2k_{B}T}{ A(q^{2} + {k_{0}^{2}}) + B(q^{4} + {k_{0}^{4}}) }. $$

(29)

Rescale the momentum q=q ^′/b changes the correlation to

$$ G_{ii}(q) = \frac{2k_{B}Tb^{2}} { A\left( q^{\prime 2} + b^{2}{k_{0}^{2}} \right) + \frac{B}{b^{2}} \left( q^{\prime 4} + b^{4}{k_{0}^{4}} \right) } $$

(30)

whose the scaling law is $G_{ii}(q) = b^{2}G_{ii}^{\prime }(q^{\prime })$, where the correlation $G_{ii}^{\prime }(q^{\prime })$ has the identical form with G _{i i} (q) but with the renormalized stretch modulus A ^′, the renormalized bending modulus B ^′, and the renormalized helix wavevector $k_{0}^{\prime }$

$$ G_{ii}^{\prime}(q^{\prime}) = \frac{2k_{B}T}{ A^{\prime}\left( q^{\prime 2} + k_{0}^{\prime 2} \right) + B^{\prime}\left( q^{\prime 4} + k_{0}^{\prime 4} \right) }. $$

(31)

To fulfill the scaling law of correlation, we choose the rescaling parameter ζ ²=2b ³, which simplifies (25) to

$$ \begin{array}{ccccc} A^{\prime} = A, \hspace{5mm} & B^{\prime} = \frac{1}{b^{2}}B, \hspace{5mm} & k_{0}^{\prime} = bk_{0}, \hspace{5mm} & F_{x}^{\prime} = \frac{1}{\sqrt{2}} b^{3/2}F_{x}, \hspace{5mm} & F_{y}^{\prime} = \frac{1}{\sqrt{2}} b^{3/2}F_{y}. \end{array} $$

(32)

which are indeed the recursion relations (17) in the main text.