We generated unsymmetric distance matrix which measure the similarity between samples.
Could you recommend to a R or python function, or a custom code to perform permanova analysis?
I would go with the R package vegan. It has all of the methods you need, including ANOSIM or ADONIS. I would also suggest following and watching The Riffomonas Project on Youtube, by Prof. Pat Schloss. He deals a lot with statistical comparison in microbial ecology (applicable in ecology in general), and how to perform it using R.
This question is about carrying out methods such as Permanova on an asymmetric distance matrix. This implies that the distance from sample (or variable) A to B is not the same as the distance from B to A. Most distance-based methods consider symmetric measures (distance A to B = B to A), so the question is not about the code you need to do a Permanova (or other) test, but whether the test is possible given your chosen distance measure. I don't think it is. So, you need to think about why you have chosen to use an asymmetric measure, what that asymmetry tells you, and whether it is important to what you want to know. If it isn't, you could average the distances A to B and B to A to get a single distance for each pair of objects (samples presumably, but could be variables), or choose another symmetric measure, and take it from there. If it is, then some other analytical approach might take you to where you want to be. I'd need to know more to say more.
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