in most cases, you will not know the effect size before running any pilot experiments. Sometimes, you don't even know if your treatment will have any effect on your system, as this actually is your research question. Sample size estimation in such cases is highly speculative, I would recommend to start a small pilot study to get an idea of data spread (assuming that variability is not altered by your treatment) and define a "biologically significant" effect, e.g. say "I would consider a difference of 20 % in the endpoint observed meaningful" (not statistically significant). From those mean differences and data spread you calculate your effect size, from which you will proceed to sample size calculation (using the appropriate table / calculator for your particular statistical test and your pre-defined acceptable type-I and type-II errors). Just to be safe, go a little higher than this number, as you get a more robust test.
Especially considering ethical animal welfare issues, do NOT go with the determined minimum numbers but always a little higher, as any random outliers may invalidate conclusive evaluation of the data, which strictly means that all your experimental animals have been sacrificed without getting a correct and statistically significant (and therefore conclusive) result. Then you may have to repeat the whole experiment, at least doubling the total numbers of mice sacrificed.