One way to do it is:
while read p; do echo "$p" done <peptides.txt
As pointed out in the comments, this has the side effects of trimming leading whitespace, interpreting backslash sequences, and skipping the last line if it's missing a terminating linefeed. If these are concerns, you can do:
while IFS="" read -r p || [ -n "$p" ] do printf '%s\n' "$p" done < peptides.txt
Exceptionally, if the loop body may read from standard input, you can open the file using a different file descriptor:
while read -u 10 p; do ... done 10<peptides.txt
Here, 10 is just an arbitrary number (different from 0, 1, 2).