Is there any way to do a power-based sample size calculation if you don't know the minimal (scientifically) important difference? I am in the life sciences field and I don't have much experience with power-based sample size calculations. I am doing a Western blot experiment and my research question is whether protein X influences the levels of protein Y during development. In this experiment, I have two groups of mice (that differ based on their genotype), Genotype A and Genotype B. These groups are further subdivided by age, so in total I have four groups of mice:

*

*Genotype A, 10 day old mice

*Genotype A, 20 day old mice

*Genotype B, 10 day old mice

*Genotype B, 20 day old mice

My outcome measure is the levels of protein Y in the brains of these mice. In this study, I am interested in seeing a difference between any pair of the 4 groups. I am using the two-way ANOVA test to analyse my data since I am interested in seeing how age and genotype in combination affects the levels of protein Y.
I want to do a sample size calculation so that my study has enough power to detect a meaningful difference in protein Y levels between these groups of mice. I am not that experienced with power-based sample size calculations but I know that you need to know the following things: null hypothesis, alternative hypothesis, statistical test (two-way ANOVA), desired type 1 error rate, desired power and minimal scientifically important difference.
My desired type 1 error rate is 0.05 and my desired power is 80%. I know my null & alternative hypotheses and I'm using a two-way ANOVA followed by Sidak's multiple comparisons post hoc test. However, the only thing that I don't know is the minimal scientifically important difference (Hm). I know that you can look at previous published studies to get an idea of Hm but there are no previously published studies similar to what I am doing. Further, this is the first time I am doing these experiments so I am not sure what Hm is myself (because I have read that you can do a pilot study to find Hm if it's not something you can find out from the literature).
I know that you can do a power-based sample size calculation using the G*Power software. However, in this software, you need to know what the minimal scientifically important difference is to calculate sample size that will give you the desired power. I know that if you don't know what the meaningful difference is, you can do a sensitivity power analysis - where you calculate the effect size you can detect with a certain number of mice. However, at this point of the experiment, this information is not useful.
I was wondering is there any way at all to do a power-based sample size calculation if you don't know what the minimal scientifically important difference is? E.g. what sample size would I need to have a power of 80%?
Any advice is appreciated.

As an aside type I error rate $\alpha$ is neither a rate nor an error.  I prefer to call it type I assertion probability.
The answer to your question is no.  But you could use slightly different wording for the parameter that needs to go into the power calculation (besides $\alpha, n, \sigma$).  Call it the effect you would be embarrassed to miss.  This may harmonize with the minimal biologic effect in many cases.
For many problems, possibly including yours, power is a less useful concept than estimating the likely information yield of the experiment.  A simple way to think of information yield is the margin of error in estimating a key quantity of interest.  This quantity may be a difference in means between two groups, and correspond to half the width of, say, a 0.95 confidence interval.  This approach is usually called planning for precision, and I delve into this in BBR.  Instead of a "difference not to miss" you have to specify an acceptable margin of error.  This is not necessarily a trivial task but is a bit easier than agreeing on a biologic effect not to miss.  And it is more useful in my view, because if you run the experiment and get p=0.12 you have no idea what to conclude, but the compatibility (confidence) interval has meaning whether you accept or reject $H_0$.
