G. Elastic fluctuations in the mesoscopic model

In order to obtain a full description of the model that includes the effects of the elastic fluctuations of the polymers, we have to split the contribution of the handles and the ssDNA, so that they can have independent lengths (see Fig. G.1).

Figure G.1: Full model. Each element is treated indepentdently.
\includegraphics[width=10cm]{figs/appendix3/fullmodel.eps}

The total energy of the system is then written as,

$\displaystyle G_\mathrm{tot}(x_{h_1},x_{s_1},x_{s_2},x_{h_2},x_b,n)$ $\displaystyle =$ $\displaystyle G_{h_1}(x_{h_1})+G_{s_1}(x_{s_1},n)+G_\mathrm{DNA}(n)$  
  $\displaystyle +$ $\displaystyle G_{s_2}(x_{s_2},n)+G_{h_2}(x_{h_2})+E_b(x_b)$  

where the explicit dependencies of the handles are given by Eq. F.2; either Eq. F.2 or Eq. F.5 give the dependency of the ssDNA; Eq. 3.17 gives the free energy of the duplex and Eq. 3.16 gives the energy of the bead in the optical trap. The total extension of the system is the control parameter and it can be written as

$\displaystyle x_\mathrm{tot}=x_{h_1}+x_{s_1}+\phi_\mathrm{DNA}+x_{s_2}+x_{h_2}+\phi/2+x_b$ (G.1)

where $ \phi_\mathrm{DNA}$ is the diameter of the DNA duplex and $ \phi$ is the diameter of the bead. The partition function can be calculated by summing over all the possible states of the system according to

$\displaystyle Z(x_\mathrm{tot})=\sum_{n=0}^N {\int_{\mathbb{R}^5\ge 0} \!\!\! e...
...ight\vert _{x_\mathrm{tot}}} \cdot dx_{h_1} dx_{s_1} dx_{s_2} dx_{h_2} dx_{b} }$ (G.2)

where the integral is extended over all positive values of the extensions $ x_{h_1},x_{s_1},x_{s_2},x_{h_2}$ and $ x_{b}$; provided that Eq. G.2 is fulfilled. Now, the calculation of this integral is quite complex because the extensions in Eq. G.3 are restricted by Eq. G.2, so they are not independent variables. Our goal is to write the total free energy of the system in terms of independent variables that can be integrated separately. So we can define a new set of positively defined variables $ x_1$, $ x_2$, $ x_3$ and $ x_4$ (see Fig. G.1) that indicate the position of the ends of the elastic elements, which are related to the extensions according to the following relations:

$\displaystyle \left\{
 \begin{array}{rcl}
 x_{h_1}&=&x_1 \\ 
 x_{s_1}&=&x_2-x_1...
... 
 x_{h_2}&=&x_4-\phi/2 - x_3 \\ 
 x_b&=&x_\mathrm{tot}-x_4
 \end{array}\right.$ (G.3)

and allow us to rewrite Eq. G.1 as
$\displaystyle G_\mathrm{tot}(x_1,x_2,x_3,x_4,x_\mathrm{tot},n)$ $\displaystyle =$ $\displaystyle G_{h_1}(x_1)+G_{s_1}(x_2-x_1,n)+$  
  $\displaystyle +$ $\displaystyle G_\mathrm{DNA}(n)+G_{s_2}(x_3-x_2-\phi_\mathrm{DNA},n)+$  
  $\displaystyle +$ $\displaystyle G_{h_2}(x_4-\phi/2-x_3) + E_b(x_\mathrm{tot}-x_4)$ (G.4)

Now the total energy of the system can be calculated for different extensions of the elements without affecting the total extension of the system. For instance, we can give different values to $ x_1$ that modify the extensions of $ x_{h_1}$ and $ x_{s_1}$ without changing the value of $ x_\mathrm {tot}$. Now, the partition function can be calculated by integrating over the new variables of the elements according to

$\displaystyle Z(x_\mathrm{tot})= \sum_{n=0}^N \int_{\mathbb{R}^4} e^{\displayst...
... G_\mathrm{tot}(x_1,x_2,x_3,x_4,x_\mathrm{tot},n)} \cdot dx_1\,dx_2\,dx_3\,dx_4$ (G.5)

where the integral is extended over all values of $ -\infty<x_i<+\infty$, $ i=1,\dots,4$ while keeping the value of $ x_\mathrm {tot}$ constant. Note that Eqs. G.4 define a set of limits of integration for the new variables that do not coincide with the ones used in Eq. G.6. Nevertheless, we extend the limits of integration of the newly defined variables over all the space. We can do this because the contribution to the integral vanishes for large extensions, due to the exponential dependence of the energy. The advantage of such approximation is that we will have to perform Gaussian integrals, which are easy to calculate when the limits of integration expand over all the space.

For each value of $ x_\mathrm {tot}$ and $ n$, there is always a combination of $ (x_1,x_2,x_3,x_4)$ that minimizes the total energy in Eq. G.5. It is convenient to define this global minimum as $ G_\mathrm{min}(x_\mathrm{tot},n)$ because it allows to perform the integral by using a saddle-point approximation. If we define $ \vec{x}=(x_1,x_2,x_3,x_4)$ as the vector that contains all the variables, we can perform a Taylor expansion of Eq. G.5 around the minimum $ \vec{x}_\mathrm{m}$ of $ G_\mathrm{tot}(\vec{x},x_\mathrm{tot},n)$ up to order $ \mathcal{O}(\vec{x}-\vec{x}_\mathrm{min})^2$ according to,

$\displaystyle G_\mathrm{tot}(\vec{x},x_\mathrm{tot},n)$ $\displaystyle \simeq$ $\displaystyle G_\mathrm{min}(x_\mathrm{tot},n)+\left. \vec{\nabla}_{\vec{x}}\, ...
...ight\vert _{\vec{x}_\mathrm{min}}\!\!\!\!\!\cdot (\vec{x}-\vec{x}_\mathrm{min})$  
    $\displaystyle +\frac{1}{2}\,(\vec{x}-\vec{x}_\mathrm{min})^T \cdot \mathbf{H}\B...
...,n)\Big)_{\vec{x}_\mathrm{min}} \!\!\!\!\! \cdot (\vec{x}-\vec{x}_\mathrm{min})$  

where $ (\vec{x}-\vec{x}_\mathrm{min})^T$ stands for the transposed vector and $ \mathbf{H}$ is the Hessian matrix calculated at fixed $ x_\mathrm {tot}$ and $ n$. Formally, the Hessian matrix is a quadratic form that acts over the vector $ (\vec{x}-\vec{x}_\mathrm{min})$ and its transposed $ (\vec{x}-\vec{x}_\mathrm{min})^T$ and returns the change in $ G_\mathrm{tot}$. Note that the variables of the function are $ \vec{x}$, while $ x_\mathrm {tot}$ and $ n$ are fixed parameters. Since the Taylor expansion has been performed around the minimum, the gradient (i.e., first derivative) of the function vanishes and the second term in the right side of Eq. G.7 does not contribute.

In order to obtain the Hessian matrix, we have to calculate all the second order derivatives of Eq. G.1. The first derivatives of elastic energy terms give forces and the second derivatives give stiffnesses. So for the first derivatives of Eq. G.5 we can write

$\displaystyle \left.\frac{\partial\,G_\mathrm{tot}(\vec{x})}{\partial x_1}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial \left(G_{h_1}(x_1)+G_{s_1}(x_2-x_1)\right)}{\partial x_1}$  
  $\displaystyle =$ $\displaystyle f_{h_1}(x_1)-f_{s_1}(x_2-x_1)$  
       
$\displaystyle \left.\frac{\partial\,G_\mathrm{tot}(\vec{x})}{\partial x_2}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial \left(G_{s_1}(x_2-x_1)+G_{s_2}(x_3-x_2-\phi_\mathrm{DNA})\right)}{\partial x_2}$  
  $\displaystyle =$ $\displaystyle f_{s_1}(x_2-x_1)-f_{s_2}(x_3-x_2-\phi_\mathrm{DNA})$  
       
$\displaystyle \left.\frac{\partial\,G_\mathrm{tot}(\vec{x})}{\partial x_3}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial \left(G_{s_2}(x_3-x_2-\phi_\mathrm{DNA})+G_{h_2}(x_4- \phi/2 - x_3)\right)}{\partial x_3}$  
  $\displaystyle =$ $\displaystyle f_{s_2}(x_3-x_2-\phi_\mathrm{DNA})-f_{h_2}(x_4- \phi/2 - x_3)$  
       
$\displaystyle \left.\frac{\partial\,G_\mathrm{tot}(\vec{x})}{\partial x_4}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial \left(G_{h_2}(x_4-\phi/2 - x_3) + E_b(x_\mathrm{tot}-x_4)\right)}{\partial x_4}$  
  $\displaystyle =$ $\displaystyle f_{h_2}(x_4-\phi/2-x_3) - f_b(x_\mathrm{tot}-x_4)$ (G.6)

where $ f_i(x_i)$ $ i=h_1,s_1,h_2,s_2$ are the FEC of the elastic elements and the explicit dependencies of $ G_\mathrm{tot}$ on $ x_\mathrm {tot}$ and $ n$ have not been written. The components of the Hessian matrix will be given by

$\displaystyle \textbf{H}\left(G_\mathrm{tot}(\vec{x})\right)=
 \left(
 \renewed...
... & \dfrac{\partial^2 G_\mathrm{tot} }{\partial x^2_4} \\ 
 \end{array}
 \right)$ (G.7)

which can be computed from Eqs. G.8 according to
$\displaystyle h_{11}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_1^2}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_1} \left( \frac{\partial G_\mathrm{tot}}{\partial x_1} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_1} \Big( f_{h_1}(x_1)-f_{s_1}(x_2-x_1) \Big)$  
  $\displaystyle =$ $\displaystyle k_{h_1}(x_1)+k_{s_1}(x_2-x_1)$  
  $\displaystyle =$ $\displaystyle k_{h_1}(x_{h_1})+k_{s_1}(x_{s_1})$  
       
$\displaystyle h_{21}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_2 \partial x_1}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_2} \left( \frac{\partial G_\mathrm{tot}}{\partial x_1} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_2} \Big( f_{h_1}(x_1)-f_{s_1}(x_2-x_1) \Big)$  
  $\displaystyle =$ $\displaystyle -k_{s_1}(x_2-x_1)$  
  $\displaystyle =$ $\displaystyle - k_{s_1}(x_{s_1})$  
       
$\displaystyle h_{22}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_2^2}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_2} \left( \frac{\partial G_\mathrm{tot}}{\partial x_2} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_2} \Big( f_{s_1}(x_2-x_1)-f_{s_2}(x_3-x_2-\phi_\mathrm{DNA}) \Big)$  
  $\displaystyle =$ $\displaystyle k_{s_1}(x_2-x_1)+k_{s_2}(x_3-x_2-\phi_\mathrm{DNA})$  
  $\displaystyle =$ $\displaystyle k_{s_1}(x_{s_1})+k_{s_2}(x_{s_2})$  
       
$\displaystyle h_{12}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_1 \partial x_2}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_1} \left( \frac{\partial G_\mathrm{tot}}{\partial x_2} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_1} \Big( f_{s_1}(x_2-x_1)-f_{s_2}(x_3-x_2-\phi_\mathrm{DNA}) \Big)$  
  $\displaystyle =$ $\displaystyle -k_{s_1}(x_2-x_1)$  
  $\displaystyle =$ $\displaystyle - k_{s_1}(x_{s_1})$  
       
$\displaystyle h_{32}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_3 \partial x_2}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_3} \left( \frac{\partial G_\mathrm{tot}}{\partial x_2} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_3} \Big( f_{s_1}(x_2-x_1)-f_{s_2}(x_3-x_2-\phi_\mathrm{DNA}) \Big)$  
  $\displaystyle =$ $\displaystyle -k_{s_2}(x_3-x_2-\phi_\mathrm{DNA})$  
  $\displaystyle =$ $\displaystyle -k_{s_2}(x_{s_2})$  
       
$\displaystyle h_{33}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_3^2}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_3} \left( \frac{\partial G_\mathrm{tot}}{\partial x_3} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_3} \Big( f_{s_2}(x_3-x_2-\phi_\mathrm{DNA})-f_{h_2}(x_4- \phi/2 - x_3) \Big)$  
  $\displaystyle =$ $\displaystyle k_{s_2}(x_3-x_2-\phi_\mathrm{DNA})+k_{h_2}(x_4- \phi/2 - x_3)$  
  $\displaystyle =$ $\displaystyle k_{s_2}(x_{s_2})+k_{h_2}(x_{h_2})$  
       
$\displaystyle h_{23}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_2 \partial x_3}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_2} \left( \frac{\partial G_\mathrm{tot}}{\partial x_3} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_2} \Big( f_{s_2}(x_3-x_2-\phi_\mathrm{DNA})-f_{h_2}(x_4- \phi/2 - x_3) \Big)$  
  $\displaystyle =$ $\displaystyle -k_{s_2}(x_3-x_2-\phi_\mathrm{DNA})$  
  $\displaystyle =$ $\displaystyle -k_{s_2}(x_{s_2})$  
       
$\displaystyle h_{43}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_4 \partial x_3}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_4} \left( \frac{\partial G_\mathrm{tot}}{\partial x_3} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_4} \Big( f_{s_2}(x_3-x_2-\phi_\mathrm{DNA})-f_{h_2}(x_4- \phi/2 - x_3) \Big)$  
  $\displaystyle =$ $\displaystyle -k_{h_2}(x_4- \phi/2 - x_3)$  
  $\displaystyle =$ $\displaystyle -k_{h_2}(x_{h_2})$  
       
$\displaystyle h_{44}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_4^2}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_4} \left( \frac{\partial G_\mathrm{tot}}{\partial x_4} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_4} \Big( f_{h_2}(x_4-\phi/2-x_3) - f_b(x_\mathrm{tot}-x_4) \Big)$  
  $\displaystyle =$ $\displaystyle k_{h_2}(x_4-\phi/2-x_3) + k_b(x_\mathrm{tot}-x_4)$  
  $\displaystyle =$ $\displaystyle k_{h_2}(x_{h_2}) + k_b(x_b)$  
       
$\displaystyle h_{34}=\left. \frac{\partial^2 G_\mathrm{tot}(\vec{x})}{\partial x_3 \partial x_4}\right\vert _{x_\mathrm{tot},n}$ $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_3} \left( \frac{\partial G_\mathrm{tot}}{\partial x_3} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\partial }{\partial x_3} \Big( f_{h_2}(x_4-\phi/2-x_3) - f_b(x_\mathrm{tot}-x_4) \Big)$  
  $\displaystyle =$ $\displaystyle -k_{h_2}(x_4- \phi/2 - x_3)$  
  $\displaystyle =$ $\displaystyle -k_{h_2}(x_{h_2})$ (G.8)

where $ k_i(x_i)$ ( $ i=h_1,s_1,s_2,h_2,b$) are the stiffnesses of the elastic components (handles, ssDNA and optical trap) and the variables $ x_i$ ( $ i=1,\dots,4$) have been written in terms of the extensions again. All the other components of the Hessian matrix ( $ h_{13},h_{14},h_{24},h_{31},h_{41},h_{42}$) vanish because the first derivative does not depend on the the variable that is being derivated in second order. For instance, $ h_{14}=0$ because $ \frac{\partial G_\mathrm{tot}(\vec{x})}{\partial x_4}$ does not depend on $ x_1$. Note that all second derivatives are symmetric ( $ h_{ij}=h_{ji}$), as expected in a function that has continuous second partial derivatives. Now the Hessian matrix can be written as

$\displaystyle \textbf{H}\left(G_\mathrm{tot}(\vec{x})\right)=
 \left(
 \begin{a...
...h_{32} & h_{33} & h_{34} \\ 
 0 & 0 & h_{43} & h_{44} \\ 
 \end{array}
 \right)$ (G.9)

and its determinant is equal to
$\displaystyle \det \textbf{H}\left(G_\mathrm{tot}(\vec{x})\right)$ $\displaystyle =$ $\displaystyle h_{11} h_{22} h_{33} h_{44} - h_{11} h_{22} h_{34} h_{43} - h_{11} h_{23} h_{32} h_{44}$  
    $\displaystyle - h_{12} h_{21} h_{33} h_{44} + h_{12} h_{21} h_{34} h_{43}$ (G.10)

which will be necessary to compute the integral. At this point it is important to remember that the components of the Hessian matrix (and so the determinant) depend on $ x_\mathrm {tot}$ and $ n$ and the calculation of the stiffnesses (i.e., the second order derivatives) is performed at fixed values of $ x_\mathrm {tot}$ and $ n$.

Now we recall the Taylor expansion written in Eq. G.7 and we introduce it into the calculation of the partition function in Eq. G.6 to obtain the following expression,

$\displaystyle Z(x_\mathrm{tot})$ $\displaystyle =$ $\displaystyle \sum_{n=0}^N \int_{\mathbb{R}^4} e^{\displaystyle -\beta G_\mathrm{tot}(x_1,x_2,x_3,x_4,x_\mathrm{tot},n)} \cdot dx_1\,dx_2\,dx_3\,dx_4$  
  $\displaystyle \simeq$ $\displaystyle \sum_{n=0}^N \int_{\mathbb{R}^4} \, e^{\displaystyle -\beta \Big(...
...)^T \cdot \mathbf{H} \cdot (\vec{x}-\vec{x}_\mathrm{min}) \Big)} \cdot d\vec{x}$  

where the gradient of $ G_\mathrm{tot}(\vec{x})$ vanishes because we are expanding around the minimum. Now we can perform the integral of the quadratic contribution according to
$\displaystyle Z(x_\mathrm{tot})$ $\displaystyle =$ $\displaystyle \sum_{n=0}^N e^{\displaystyle -\beta G_\mathrm{min}(x_\mathrm{tot...
...m{min})^T \cdot \mathbf{H} \cdot (\vec{x}-\vec{x}_\mathrm{min})} \cdot d\vec{x}$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^N e^{\displaystyle -\beta G_\mathrm{min}(x_\mathrm{tot...
...laystyle -\tfrac{\beta}{2} \: \vec{y}^{\,T}\!\mathbf{H} \vec{y}} \cdot d\vec{y}$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^N e^{\displaystyle -\beta G_\mathrm{min}(x_\mathrm{tot...
...t{\left(\frac{2\pi}{\beta}\right)^4\frac{1}{\det \textbf{H}(x_\mathrm{tot},n)}}$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^N \left(\frac{2\pi}{\beta}\right)^2 \big[\det \textbf{...
...tot},n)\big]^{-1/2} \;e^{\displaystyle -\beta G_\mathrm{min}(x_\mathrm{tot},n)}$ (G.11)

where we have applied a change of variable $ (\vec{x}-\vec{x}_\mathrm{min})\rightarrow \vec{y}$ that keeps the same integration limits and allows to perform the Gaussian integration by computing the determinant of the Hessian matrix.

So, in order to calculate the partition function of the whole model we have proceed as follows:

  1. Fix a value of $ x_\mathrm {tot}$ at which $ Z(x_\mathrm{tot})$ will be calculated.
  2. For this fixed value of $ x_\mathrm {tot}$, we have to find the total energy for all the values of $ n$. It requires to solve the transcendental equation G.2 for fixed values of $ x_\mathrm {tot}$ and $ n$, find the extensions of all the elastic elements and introduce them into Eq. G.1.
  3. For each value of $ n$ (at the same fixed value of $ x_\mathrm {tot}$), we have to calculate the Hessian matrix according to Eqs. G.10 and compute its determinant, given by Eq. G.12.
  4. Once we have all the values of $ G_\mathrm{min}(x_\mathrm{tot},n)$ and $ \det \textbf{H}(x_\mathrm{tot},n)$ from steps 2 and 3 respectively for all values of $ n$, we have to multiply them and sum over $ n$ according to Eq. G.13.
  5. Go back to step 1 to calculate $ Z(x_\mathrm{tot})$ at another value of $ x_\mathrm {tot}$.

Once the partition function is calculated at all desired values of $ x_\mathrm {tot}$, the equation of state is obtained after derivating it according to Eq. 3.27,

$\displaystyle f(x_\mathrm{tot})=-k_B T\frac{\partial \ln Z(x_\mathrm{tot})}{\partial x_\mathrm{tot}}$ (G.12)

Figure G.2 shows the correction introduced in the calculation of the FDC when considering the elasticity of the elements that constitute the whole model. Note that the differences are less than 0.05 pN. Therefore the effort to compute all the second derivatives of the Hessian matrix does not compensate the improvement in the computation of the FDC, which is lower than the experimental resolution of the experiments. However, this effect is important when the length of the handles is longer and they no longer behave like rigid rods.

Figure G.2: FDCs for the simplified and full models. Red curve shows the simplified model described in Sec. 3.4.1. Blue curve shows the full model described here in Appendix G.
\includegraphics[width=\textwidth]{figs/appendix3/hessnohess.eps}

JM Huguet 2014-02-12