It is pretty straightforward to run mixed-effects models in R using lme4 or nlme packages.

You can also try limma package which is specifically designed for gene expression analysis including RNAseq.

As Maxim has already pointed out, there are several options and implementations of the functionality in R alone and many more, if you prefer Python or BioGo or such.

However, I assume you would like to know how to specify the models' terms and

Post hoc test the fitted models afterwards.

In this case, you might find chapters 3 + 4 of the edgeR - User Guide helpful as a starter:

http://bioconductor.org/packages/release/bioc/vignettes/edgeR/inst/doc/edgeRUsersGuide.pdf

For Post Hoc Tests in R you could try the emmeans package or (as a starter) test at least your linear hypotheses with glht():

https://cran.r-project.org/web/packages/emmeans/vignettes/basics.html

https://stat.ethz.ch/pipermail/r-help/2012-January/299623.html

limma was developed for microarrays analysis and is not really dedicated to RNA-seq, even if it contains a dedicated module to it. I would recommend to use either edgeR or DESeq2 on raw read counts and to introduce your mix linear models according to the relevant package manual / user guide. Both packages, with detailed help are available on bioconductor and are widely used tools. Usually, you need to introduce a so-called "formula", under a form like ~factor1+factor2, ~factor1:factor2 or ~factor1*factor2, depending if you want to analyze the effect of your factors independently, their interaction or both. Good luck!

You can try to use an example here in our repository https://github.com/uhkniazi/BRC_SupernumeraryTooth_Gui_PID_19/blob/5d1fb0f56b0bc107c5c297d8fc0f2a57563a2e31/10_deAnalysis.R#L102

However the main analysis is done in Stan. There is a more detailed example here https://laplacebayes.wordpress.com/2018/02/06/high-dimensional-data-hierarchical-regression/

You have to consider that negative binomial GLMs may not converge very well if your sample sizes are small and you are using lmer. That's why we tend to use Monte Carlo methods using Stan.

Good luck

How many concentrations do you have? And how many weights? In other words, how many levels has each factor?

It appears to me that concentration and weight might be continuous variables, and they might be entered into the model as such, and not as factors. In this case it might not be necessary to decide whether they are random or fixed factors.