Scaling of the average crossing number in equilateral random walks, knots and proteins

We compare here the scaling behaviour of the mean average crossing number 〈ACN〉 of equilateral random walks in linear and closed form with the corresponding scaling observed in natural protein trajectories. We have shown recently that the scaling of 〈ACN〉 of equilateral random walks of length n follows the relation 〈ACN〉 = 3 16 n ln n + bn and that a similar result holds for equilateral random polygons [13]. Furthermore, our earlier numerical studies indicated that when random polygons of length n are divided into individual knot types, the 〈ACN(K)〉 for each knot type K can be described by a function of the form 〈ACN(K)〉 = a(n− n0) ln(n− n0) + b(n− n0) + c where a, b and c are constants depending on K and n0 is the minimal number of segments required to form K [13]. Here we analyze in addition natural protein structures and observe that the relation 〈ACN〉 = 3 16 n ln n + bn also describes accurately the scaling of 〈ACN〉 of protein backbones.