The calculation of the FDC has to be performed numerically due to the length of the DNA sequence. The computation of the DNA energy (Eq. 3.17) is a simple sum of terms that extends over all the base pairs of the molecular construct. So this large summation must be done numerically. Apart from that, the exponential terms that enter the partition function have a wide range of orders of magnitude and they must be treated correctly.
The partition function is a function of the distance (i.e., the total extension) , which is a variable that has to be discretized in order to calculate the value of at each position. The distance is divided into equidistant points , separated by a distance (see Fig. H.1). A value of nm is enough for our calculations. The details of the calculated FDC are missed for higher values of and lower values of it do not improve the calculation.
Now, for each value of we have to find the minimum of Eq. 3.22 with respect to . In order to do this, we have to solve Eq. 3.23 for all values of according to , compute the energies and get the value that minimizes the energy at fixed . Equation 3.23 is a trascendental equation and it can be solved numerically by using the Newton's method. This method gives the equilibrium force of the system after few interations (no more than 10 to obtain a numerical estimation of with a relative error smaller than 10) of the following expression
Once we have the combination of values () we can now calculate the minimum energy
according to Eq. 3.22. The discretized Eq. 3.27 can be written as
This problem can be fixed by taking advantage of the calculated value of
. The idea consists in rewriting the partition function with all the energies referred to this state. So we have,
To sum up, the numerical calculation of the FDC requires the following steps,
JM Huguet 2014-02-12