I would be very happy if someone can help me to find the answers for the following related questions.
- Can one exon have many stop codons?
- Can protein synthesis happen, if the stop codon is at the beginning of the exon?
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An exon can have multiple stop codons but the first codon will terminate the ORF. The remainder of the exon will be a part of the 3'UTR.
However, there are some cases in which stop codon readthrough happens . In these cases an internal stop codon does not terminate the translation and the ribosome reads through it. During this process an aminoacyl-tRNA successfully competes with the release factor. This is quite evident in the case of selenocysteine and pyrollysine- tRNAs which bind to the
UAG (amber) stop codon in certain organisms [2,3] (this is also referred to as amber suppression). The exact mechanisms that affect stop codon readthrough are not elucidated but features in the mRNAs are likely to play a role in this. In case of amber suppression, the relative rarity of the
UAG stop codon and lower concentrations of release factor-1, can be major factors.
I don't know what you mean by "beginning of the exon" but there are short ORFs (< 100 codons) which can code for small peptides . Small peptide as small as 6 amino acid residues has been reported but most of the known small peptides are ~30-100 residues long.
 Loughran, Gary, et al. "Evidence of efficient stop codon readthrough in four mammalian genes." Nucleic acids research 42.14 (2014): 8928-8938.
 Agafonov, Dmitry E., et al. "Efficient suppression of the amber codon in E. coli in vitro translation system." FEBS letters 579.10 (2005): 2156-2160.
 Wikipedia: Expanded genetic code
 Andrews, Shea J., and Joseph A. Rothnagel. "Emerging evidence for functional peptides encoded by short open reading frames." Nature Reviews Genetics 15.3 (2014): 193-204.