3.4.1 Controlled position

The position of the center of the optical trap is the control parameter that characterizes the controlled position ensemble. In this ensemble, the position of the trap is fixed and the force applied to the ends of the molecular construct fluctuates. The equation of state of the system in this ensemble can be experimentally reproduced by measuring the average force exerted on the molecular construct for each fixed position of the optical trap. The natural thermodynamic potential of this ensemble is the Helmholtz free energy, which is expressed in terms of the extensive variable (the distance) and the temperature. Therefore the energetic contributions of the model have to be expressed in terms of distances (or extensions).

The description of the molecule is split into three different parts: the handles, the open base pairs (which are in the form of single stranded DNA) and the closed base-pairs (see Fig. 3.15). We also have to take into account the effect of the optical trap. The bead in the optical trap is modeled by a Hookean spring,

$\displaystyle f=kx_{b}$ (3.15)

where $ f$ is the force applied, $ k$ is the measured stiffness of the optical trap and $ x_{b}$ is the elongation of the bead from the center of the trap. So, the potential energy of the bead in the optical trap $ E_{b}$ is described by a harmonic potential

$\displaystyle E_{b}=\frac{1}{2}kx^2_{b}~.$ (3.16)

Figure 3.15: Mesoscopic model. Each element is represented using a different color. A sketch of the free energy contribution is also shown for each element.

We use the NN model to describe the free energy of formation of the DNA duplex. The free energy required to open $ n$ base pairs is given by the sum of the free energies required to open each consecutive nearest-neighbor pair,

$\displaystyle G_\mathrm{DNA}(n)=-\sum_{i=0}^n \epsilon_i$ (3.17)

where $ G_\mathrm{DNA}(n)$ is the free energy of the hairpin when $ n$ bps are disrupted and $ \epsilon _i$ is the free energy of formation of the bp $ i$. The values of $ \epsilon _i$ are negatively defined (because the base-pairing is a spontaneous reaction) and they are identical to $ \Delta g_i$ (see Eq. 3.5 for a definition of $ \Delta g^0_i$ in standard conditions). The resulting function $ G_\mathrm{DNA}(n)$ is positively defined and monotonically increasing with $ n$ (see Fig. 3.15). So, the energy is minimum and equal to 0 when all the base pairs are closed ( $ G_\mathrm{DNA}(0)=0$) and it is maximum when all base pairs are open ( $ G_\mathrm{DNA}(N)=\sum_{i=1}^N\epsilon_i)$. The free energy of the duplex depends on the sequence of base pairs. An extra free energy contribution is included in the model to account for the disruption of the end loop when all the base pairs are open (i.e., $ n=N$).

Elastic models for polymers are used to describe the elasticity of the handles and the ssDNA released during the unzipping process. The handles are dsDNA and they are modeled using the force vs. extension curve of a Worm-Like Chain (WLC)

$\displaystyle f_h(x_{h})=\frac{k_B T}{4 l_p}\left( \left( 1- \frac{x_{h}}{L_0}\right)^{-2} -1 + 4\frac{x_{h}}{L_0} \right)$ (3.18)

where $ k_B$ is the Boltzmann constant and $ T$ is the temperature, $ l_p$ is the persistence length and $ L_0$ is the contour length. The elastic free energy of the handles ( $ G_{h}(x_{h})$) is obtained by integrating the previous expression (see Appendix F for further details) according to

$\displaystyle G_h(x_h)=\int_0^{x_h} f_h(x')\,dx'$ (3.19)

where $ x'$ is a dummy variable. The ssDNA is modeled using either a WLC or a Freely-Jointed Chain (FJC) model. Depending on the salt concentration of the experiment, one model or the other describes better the elastic response of the ssDNA. In the case of the FJC model, the following equation gives the extension vs. force curve,

$\displaystyle x_{s}(f,n)=nd\cdot \left( \coth \left(\frac{bf}{k_B T}\right) - \frac{k_B T}{bf}\right)$ (3.20)

where $ k_B$ is the Boltzmann constant and $ T$ is the temperature, $ b$ is the Kuhn length, $ n$ is the number of bases and $ d$ is the interphosphate distance of the ssDNA. The product $ L_0=n\times d$ is the contour length of the molecule. It is convenient to write the explicit dependence on $ n$ in Eq. 3.20 because $ n$ is not a parameter but a variable that changes during unzipping. Again, the elastic free energy of the ssDNA is obtained by integrating the force versus molecular extension. However, in Eq. 3.20 $ f$ is the independent variable so the calculation of $ G_{s}(x_{s},n)$ requires the inversion of $ x_s(f,n)$. It can be avoided by performing an integration by parts (see Appendix F) according to

$\displaystyle G_s(x_s,n)=\int_0^{x_s} f_s(x',n)\,dx'=f\cdot x_s(f,n)-\int_0^{f}x_s(f',n)\,df'$ (3.21)

where $ f'$ is a dummy variable and the limits of integration are related by $ x_s=x_s(f,n)$. The total free energy of the global system ( $ G_\mathrm{tot}$) is given by the sum of all free energy contributions,

$\displaystyle G_\mathrm{tot}(x_\mathrm{tot},n)=E_{b}(x_{b})+2G_{h}(x_{h})+2G_{s}(x_{s},n)+G_\mathrm{DNA}(n)$ (3.22)

where $ x_\mathrm {tot}$ is the total extension of the system. The energy of the handles and the ssDNA has to be counted twice because there are two handles and two fragments of released ssDNA. Eventually, the total free energy of the system is completely determined by the number of open base pairs and the total distance ( $ x_\mathrm {tot}$), which is given the by sum of the extensions of all the elements

$\displaystyle x_\mathrm{tot}(f,n)=x_{b}(f)+2x_{h}(f)+2x_{s}(f,n)$ (3.23)

where $ x_b$ and $ x_s$ are obtained directly from Eq. 3.15 and 3.20 respectively and $ x_h$ requires to invert Eq. 3.18 numerically. The extension of the elastic elements only depend on the force $ f$ applied to the ends of the molecular construct. By doing this we are implicitly assuming that $ x_b$, $ x_h$ and $ x_s$ are in local equilibrium and they do not fluctuate. A more accurate calculation requires to let the forces applied to the ends of each element fluctuate. However, such calculation is more complex and it does not introduce any appreciable difference in the final result. Appendix G discusses this issue in more detail.

Besides, the extension of the ssDNA in Eq. 3.23 depends on the number of open base pairs $ n$. Additionally, constant terms might also contribute to the total extension of the system (e.g., the bead diameters or the width of the DNA duplex). The thermodynamic properties of the model are not affected if these are neglected, though.

In the end, the calculation of the total energy at a given total extension $ x_\mathrm {tot}$ and number of open base pairs $ n$ requires to solve $ f$ in Eq. 3.23 by using Eqs. 3.15, 3.18 and 3.20. Once the equilibrium force of the system is known, it can be used to recover the extension of each element and calculate the total energy of the system. Therefore, Eqs. 3.22 and 3.23 form a system of two coupled equations that determine the force and the total energy of the system.

JM Huguet 2014-02-12