  Geometry and topology are the main factors that determine the functional
properties of proteins. In this work, we show how to use the Gauss linking
integral (GLN) in the form of a matrix diagram - for a pair of a loop and a
tail - to study both the geometry and topology of proteins with closed loops
e.g. lassos. We show that the GLN method is a significantly faster technique to
detect entanglement in lasso proteins in comparison with other methods. Based
on the GLN technique, we conduct comprehensive analysis of all proteins
deposited in the PDB and compare it to the statistical properties of the
polymers. We found that there are significantly more lassos with negative
crossings than those with positive ones in proteins, the average value of
maxGLN (maximal GLN between loop and pieces of tail) depends logarithmically on
the length of a tail similarly as in the polymers. Next, we show the how high
and low GLN values correlate with the internal exibility of proteins, and how
the GLN in the form of a matrix diagram can be used to study folding and
unfolding routes. Finally, we discuss how the GLN method can be applied to
study entanglement between two structures none of which are closed loops. Since
this approach is much faster than other linking invariants, the next step will
be evaluation of lassos in much longer molecules such as RNA or loops in a
single chromosome.
