There are numerous bioinformatics tools which predict the targets of a given miRNA. Is there any tool/ software to find the microRNAs regulating a given set of known genes/ mRNAs?
BLAST is not enough. Applications for miRNA target prediction are not simply sequence alignment tools. miRNA target prediction algorithms take into account the seed sequence, and compensatory binding-sites. Also the majority of them use the delta-G function of the RNA-RNA hybrid to rank among solutions. The most advanced applications are also able to consider the potential secondary structures of the target mRNA which may prevent miRNA binding.
As Francisco pointed out, BLAST may not be enough. Also, I am not doing it for a single gene. There are a set of genes for which, I need to find the miRNAs. I also used miRTarbase to find the experimentally validated mRNA-miRNA interactions.
miRNA seed region which is complementary to mRNA can be 2-8 nucleotides in length. So some times two nucleotides complementarity can repress the translation of a target mRNA. So I dont think that BLAST is the proper way to find targets of an miRNA and vice versa. Other parameters are also accountable in target prediction. The thermodynamic stability of the mRNA and miRNA duplex along with probability of the formation of secondary structure surrounding the miRNA etc., So it is better to use the tools in target prediction which can give you the targets based on several parameters.
Just to provide another answer for anyone looking at this in 2019; a handy tool I use is from DIANA: http://diana.imis.athena-innovation.gr/DianaTools/index.php?r=microT_CDS
You can paste in your list of genes and it will return provide a resulting predicted list of microRNAs.
The current work mode is to a) predict miRNAs and then b) predict their targets. If you use for example DIANA, that has been done for you (maybe not as you would like since your gene or its targeting miRNAs may have been filtered). Then you can move from target to miRNA. Of course databases such as TarBase or miRTarBase (or others) may also allow such queries. Again, whether your particular mRNA and its targeting miRNAs are in the databases is not guaranteed.
There is no tool that given an mRNA will predict the associated miRNAs from sequence as this would present a formidable challenge: Which part of the mRNA is a target site and which is not (Try guessing without knowing the miRNA)? Trying all possible e.g.: 18-mers would be possible but it doesn't seem advisable to me (you do the calculation ...). With many putative target sites you now need to go to the list of known miRNA or the genome and find complementary sites (hence blast). These you then need to check whether they form secondary structures as expected (e.g., izMiR).
In short, the road should be walked the other direction: 1) predict all miRNAs or select a list of interesting miRNAs from e.g.: miRBase, 2) predict all their targets or select targets from a database, and 3) find your gene is in the target list (hope it is there), and then 4) extract all miRNAs that target the gene.
Most suggestions here point that direction (at least implicitly), but the original question implies the opposite direction. For the desired direction mRNA -> miRNA I am not aware of a tool and I would advise against trying to implement such a tool. Reversing the logic can be successful and that is what was suggested mostly. However, whether there are known miRNAs targeting your mRNA in available databases is not guaranteed. So you could do your own prediction using psRNATarget, RNA22, RNAHybrid, ... using all relevant (e.g.: all human mature miRNAs from miRBase) miRNAs and the mRNA (check whether UTRs are present in the sequence) in question. Filter the results somehow and you will have a list of miRNAs putatively targeting your gene. If there is no answer, you could go a step further and predict all miRNAs from the genome (human will be more than 100 million candidates so not advisable) and use the filtered predictions for targeting and see which target your RNA. It would be possible, in case you have RNA-seq results, to restrict the genome space for which to predict miRNAs using only expressed regions.
Why not try some experimental approaches such as HITS-CLIP?