3.4. Controlled position
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size effects. Following, there are two separate descriptions of the model, one for each ensemble.
Figure 3.14: Ensembles. (a) Controlled position ensemble. The total distance (xtot ) between the anchor point and the center of the optical trap is held constant. The position of the bead (xb ) in the optical trap fluctuates and so does the applied force f (k is the trap stiffness). (b) Controlled force ensemble. The force is controlled by keeping constant the position of the bead in the optical trap (xb ). In order to do so, the total distance of the system (xtot ) must be corrected by a feedback. Therefore, xtot fluctuates. This ensemble can also be implemented by applying a uniform field of force as in magnetic tweezers.
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, f = kxb (3.15) where f is the force applied, k is the measured stiffness of the optical trap and xb is the elongation of the bead from the center of the trap. So, the potential