# Using Persistence length to establish straightness of position data

In the following picture I am showing 3D position data for 4 different tracks just to illustrate how the tracks looks like.

As we can see that we have all kind of behaviors occurring from helical motion(bottom left) to somewhat Brownian motion. I need to establish a metric through which I can compare these tracks with each other to establish which one is more straighter than the other. All of these tracks have different time resolution. Here is what I have tried so far:

Persistence Length is a concept associated with polymers to calculate bending stiffness. But We can also think of it as a metric to define if track is flexible or not. Here is how I calculate persistence length for these tracks.

We can assume a bond length for these tracks. We can fit a line for the points which correspond to that particular bond length. Then we can use following formula to calculate the persistence length. $$l_{p} = \frac{\sum_{1}^{n}l_{1}.l_{n}}{l} .$$ I have taken this formula from reference 1. My issue is that it works fine in the case if track does not change direction more than 90 degrees. If this change happens in the start of the track, I get negative persistence length which does not make sense to me at all. Following figure shows how it looks in practice for the case: a) when it gives something meaningful. b) when it is negative. I completely understand why I am getting negative value but this made me think that I am not using this concept correctly. If someone can point out what is my mistake here that would be really helpful. Moreover if you have another suggestion to characterize straightness please feel free to point it out.

1: Zhang, J. Z., Peng, X. Y., Liu, S., Jiang, B. P., Ji, S. C., & Shen, X. C. (2019). The Persistence Length of Semiflexible Polymers in Lattice Monte Carlo Simulations. Polymers, 11(2), 295.

The equation you cited is not the usual definition of the persistence length, $$L_p$$, which is the decay length of the average angle between segments. Perhaps this formula is roughly approximate; I'm not going to bother reading the reference. (It certainly seems odd to privilege segment 1, and not perform any averaging.) Why not calculate a persistence length in the standard way, which would prevent any weird negative lengths?

• You mean <cos T> = exp(-l/p) . I am taking T as theta and l as length and P as persistence length here. I could never understand how is averaging performed here. Because in my system I do not have any thermal fluctuations etc. Moreover if I plot this without averaging I do not get a decay function. Dec 8 '20 at 1:52
• Moreover, I do not have polymer background and any paper I find mention one or two formulas which are hard to understand when it comes to how to perform averaging. If possible can you please elaborate on what is the standard way? Dec 8 '20 at 1:54
• The averaging could be over thermal fluctuations, but could be over segments. For any pair of segments separated by distance $l$, one can calculate $T$, and so have lots of measurements of cos($T$) for the same track. (Roughly, if the total length is 100 units, you'd have 99 values of $T$ for $l=1$, about 50 for $l=2$, etc.) Dec 8 '20 at 15:19
• For background on persistence length, I'd look at polymer physics books, or biophysics books like Physical Biology of the Cell. For more description on how to extract this from images or projections, see for example Abels et al. "Single-Molecule Measurements of the Persistence Length of Double-Stranded RNA", Biophysical Journal 88: 2737–2744 (2005), and things it cites. (I'm sure there's lots of other stuff also.) Dec 8 '20 at 15:21
• Thanks a lot for you help. Reference you provided helped me a lot. I have finally solved the problem by considering each track as different thermal fluctuation. Dec 16 '20 at 20:56