3.4.2 Controlled force

In the controlled force ensemble, the force applied to the ends of the molecular construct is the control parameter and the total extension of the system is the magnitude that fluctuates. The natural thermodynamic potential of this ensemble is the Gibbs free energy, which is expressed in terms of the intensive variable (the force) and the temperature. The model is identical to the model described previously in the controlled position ensemble, but the expressions for the energies are different. Here we will follow the same description as in the previous section. The Lengendre transform converts one thermodynamic potential into the other. Using this transformation, the energetic contributions calculated at controlled position ensemble can be converted into the suitable expressions for the controlled force ensemble.

Let $ G^X(x,T)$ be the free energy (i.e., the Helmholtz free energy) of an elastic system defined by ($ x,f,T$) in the control position ensemble (i.e., fixed position and temperature). The differential form of this energy is given by

$\displaystyle dG^X(x,T)=f\,dx-S\,dT~.$ (3.28)

Now let $ G^F(f,T)$ be the corresponding free energy (i.e., the Gibbs free energy) of the same system at the controlled force ensemble (i.e., fixed force and temperature). The relation between the two thermodynamic potentials is given by $ G^F=G^X-fx$, which allows us to write
$\displaystyle dG^F(f,T)$ $\displaystyle =$ $\displaystyle dG^X(x,T)-d\,(fx)=f\,dx-S\,dT-f\,dx-x\,df$  
  $\displaystyle =$ $\displaystyle -S\,dT-x\,df~.$ (3.29)

Now it is possible to calculate the free energy of an elastic element in the controlled force ensemble by evaluating the integral of Eq. 3.29 at constant temperature ($ dT=0$) according to,

$\displaystyle G^F(f)=-\int_0^{f}x(f')\,df'$ (3.30)

where $ G^F$ is the elastic free energy, $ f$ is the force at which the energy is evaluated, $ x(f')$ is the extension vs. force curve and $ f'$ is the integration dummy variable. Eqs. 3.16, 3.19 and 3.21 are the (Helmholtz) elastic free energies (referred to as $ G^X(x)$ in the new notation introduced in this section) of the bead, the handles and the ssDNA, respectively, in the controlled position ensemble. Now, we want to calculate the (Gibbs) free energy of each of them (i.e., $ G^F(f)$) to proceed with the calculations in the controlled force ensemble, so we will apply Eq. 3.30 on all them. From now on, we will neglect the superscript $ F$ in $ G^F$ and we will understand that $ G$ must be read as the free energy in the controlled force ensemble.

The potential energy of the bead in the optical trap is given by

$\displaystyle E_b(f)=-\frac{f^2}{2k}~.$ (3.31)

The calculation of the elastic energy of the handles involves the evaluation of the following integral $ G_h(f)=-\int_{0}^{f}x_h(f')\,df'$ which has to be integrated by parts according to

$\displaystyle G_h(f)=\int_0^{x_h} f_h(x')\,dx'-fx_h$ (3.32)

where $ f_h(x')$ is given by Eq. 3.18 and the integration limits are related by $ f=f_h(x_h)$.

In the case of the ssDNA, the free energy is directly given by

$\displaystyle G_s(f,n)=-\int_0^{f} x_s(f',n)\,df'$ (3.33)

where $ x_s(f',n)$ is given in Eq. 3.20. The total energy of the system at controlled force is given by the sum of the elastic contributions (Eqs. 3.31, 3.32, 3.33) and the formation energy of the DNA (Eq. 3.17) according to

$\displaystyle G_\mathrm{tot}(f,n)=E_b(f)+2G_h(f)+2G_s(f,n)+G_\mathrm{DNA}(n)$ (3.34)

and the total extension of the system is again given by Eq. 3.23

$\displaystyle x_\mathrm{tot}(f,n)=x_{b}(f)+2x_{h}(f)+2x_{s}(f,n)~.$ (3.35)

From these two expressions it is possible to extract the properties of the DNA unzipping at controlled force as we did in the previous section with the controlled position ensemble.

JM Huguet 2014-02-12