It seems that your experiment has a very large effect size. Despite that, I suggest not to reduce the sample size to assure more credibility in the outcomes.

I thought I could use the formula of statistical power of a sample to estimate an N/group. Couldn't I?

If you have already run 4 animals per group and did a statistical significance test, your p value is gone for that study. With traditional significance testing you cannot go an collect more data and conduct another significance test. See discussion about "stopping rules", this is a topic Bayesians often bring up as a problem with frequentist statistics (i.e., p-value people). Zoltan Dienes has cartoon on this for his book (see http://www.lifesci.sussex.ac.uk/home/Zoltan_Dienes/inference/index.htm). 

If you are saying for your next study, the effect size you find in a previous study is not what you enter into you power table/packages. The effect size is the minimum that you want your study to be able to detect (given ethical/financial/etc constraints for things like getting lots of animals). Previously found effect sizes may inform your choice, but should not be the sole determinant of this.

Checking and stopping your experiment at any time during data collection is not an honest way to do statistics. Especially with small sample sizes, your effect can be influenced by chance in one or the other direction. Luckily, in your case it just turned out to be the favoured one. So does that seem a fair way to go?

To put it differently, starting with N=1 and stopping at any time just when the result turns out to be significant in the wanted direction, is just a special case of collecting all of you data and throwing out those subjects whose effect is in the wrong direction. If you see it from this perspective,it seems clear that this is not a viable option.

There are some special cases as the sequential probability ratio test, which test after each subject and have a stopping rule.

To add to Rainer's final paragraph, some Bayesians argue that with their approach you can just keep adding data, looking at the results, and then keep adding more data of you decide you want to. There are still issues that must be dealt with, but arguable it is not necessary to declare the stopping rule prior to data collection as there is with frequentism. I think the Association for Psychological Science now asked authors about stopping rules prior to submission (I think, I'd have to check).

It is a bit confusing...

You used Tukeys test, which is designed to control a family-wise type-I error rate for a family of multiple t-tests. Now, to control a type-I error rate requires that the possibility exists to commit any sensible type-I error. Therefore one required to specify two alternative models assuming that one of these will be a good description of the world and the other will be a bad description. With regard to the type-I error, the comparison of a restricted model with an unresticted model (like "equal means" vs. "unequal means") is not a suitable pair of alternatives ("unequal means" is just a pseudo-alternative; it actually includes the "equal means" hypothesis, because theier difference is not identifiable!).

Now, if you had two alternatives to plan a study in a way that the control of the family-wise error rate would make sense at all, you would not be able to simply stop with some smaller sample size then the one required to assure you the desired error-rates.

If the experiment is to find out what the differences may be, you actually have an estimation problem and not a testing problem. This is -by the way- what Bayesian methods can handle well, because stopping rules don't matter here. The posterior density will always tell you what you ought to believe about the differences, given your prejustices and any data related to the problem and known to you. To my personal opinion, I think that Bayesian testing is just a combination of the "worst of two worlds". It undermines the acknowledement of the Bayesian philosophy that data alone can never be the basis of  dichotomous decisions, and that the relevant statements should be about the credibility of the model parameters and not on the expectation of data under some particularily restriced model.