research:howto:estimating_minimal_pbc_box_for_a_folding_simulation [Norma]

Estimating the size of a safe minimal cubic PBC box for a peptide folding simulation

Beware. What follows is primitive and possibly plain wrong.

The idea is this : can we safely reduce the size of the cell for a folding simulation ? The word 'safely' in the previous sentence must be understood in a probabilistic sense : we want the smallest cubic cell for which the probability of neighboring peptide images interacting with each other (due to PBC) is less than an arbitrarily set threshold. What follows is an attempt to do just that.

Assuming an ideal chain and using the equations from this page we have :

The average end-to-end distance is $\sqrt (N-1) * l$ where $N$ is the number of amino acids and $l = 3.8$ is the average $C\alpha - C\alpha$ distance.

The variance of this end-to-end distance along any of the orthogonal axes is $\sigma^2 = (N-1) * l^2 / 3$

If we require that the edge of the (cubic) PBC box must be longer than at least $3\sigma$ plus this average end-to-end distance, then the minimal box size $K$ is given by

$K = 3.8 * \sqrt( N - 1) + 3 * \sqrt ( (N-1) * 3.8^2 / 3 )$

which reduces to

$K = 10.381793 * \sqrt( N - 1)$

Applying the above equation gives the following estimates :

Number of peptide residues	Estimated minimal cubic PBC cell edge length (Å)
10	31.2
12	34.5
14	37.4
16	40.2
18	42.8
20	45.2
22	47.6
24	49.8
26	51.9
28	54.0
30	55.9
32	57.8
34	59.6
36	61.4