Mean-Squared Displacement (MSD) - Biophysics Query

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BuddingYeastMitosis
BuddingYeastMitosis's picture
Mean-Squared Displacement (MSD) - Biophysics Query

Greetings!

I'm a grad student at Berkeley studying microtubule binding proteins.

I'm going nuts trying to figure out whether you calculate the mean-squared displacement of a group of particles at time t by taking the average of the squares of their displacements at time t from their initial positions or by taking the average of the squares of their cumulative displacements at time t.

I hope someone understands what I mean.  For example, one particle may have moved a cumulative distance of 500 nm in one second, but it may end up only 10 nm from where it started. Would you take the square of 500 or the square of 10 then to calculate the MSD.

If possible, I would also appreciate a reference to a primary article, review, or textbook that verifies the information, as it would be useful to cite.  I have searched and searched, and I have not found a satisfactory answer.

Thanks for any help!!
Nate

Sami Tuomivaara
Sami Tuomivaara's picture
BuddingYeastMitosis,

BuddingYeastMitosis,

The problem you're describing is a good example of _random walk in one dimension_. The accepted terminology is that displacement refers to the net distance between two time points, in your example, you should use 10 nm as the displacement.

You can also justify this by thinking it by as follows: Depending on the time resolution of your camera measurement, you might get different values for the total trajectory or distance traveled by the particle. Very high resolution measurement (frequent snapshots) can record every turn and jump of the particle and thus get asymptotically very accurate total trajectory for the particle (in your example, 500 nm might be quite an underestimation). Low time resolution (at time 0 and 1 s ) however only gives you the net displacement. However, no matter what the time resolution of your measurement is, you get the 10 nm as your (net) displacement at time 1 s.

Maybe these articles help. You can find more info by googling of pubmeding random walk in one dimension...

Cheers,

BuddingYeastMitosis
BuddingYeastMitosis's picture
Very good point!  I hadn't

Very good point!  I hadn't considered that.  I understand the problem you're describing.  It's like trying to measure the length of a coastline accurately.  The more precise measurement you take, the longer the coastline will appear.  Even though, in principle, the length of the coastline should be finite, the measurement seems to grow without bound with increasingly precise measurements.

I'm glad to report that the procedure I have been employing would arrive at the "10 nm" result for the example I described, so I have been doing it correctly.

Thanks very much!

Best regards,
Nate