If they are all within the same line, that is there are no line breaks between "1." and "2." then you can iterate over the lines of the file like this:

for line in open("myfile.txt"):
    #do stuff

The line will be disposed of and overwritten at each iteration meaning you can handle large file sizes with ease. If they're not on the same line:

for line in open("myfile.txt"):
    if #regex to match start of new string
       parsed_line = line
    else:
       parsed_line += line

and the rest of your code.

Maybe I'm missing something, but why don't you just use read()'s size argument?

To read a file’s contents, call f.read(size), which reads some quantity of data and returns it as a string (in text mode) or bytes object (in binary mode). size is an optional numeric argument. When size is omitted or negative, the entire contents of the file will be read and returned; it’s your problem if the file is twice as large as your machine’s memory. Otherwise, at most size bytes are read and returned.

chunksize = 1000
with open('hugefile.log', 'r') as f:
    while True:
        read_data = f.read(chunksize)
        if not read_data:
            break # done
        print(read_data)

It's not clear when do you want your chunk to end, tho - when it encounters a '>' at a beginning of a line or anywhere in the line, so I'll assume the first scenario:

chunk = []
with open("your_large_file.ext", "r") as f:
    for _ in xrange(4):  # skip 4 lines, use range() on Python 3.x instead
        next(f)
    for line in f:
        if chunk and line.startswith(">"):  # break on > if we're already collecting a chunk
            break
        chunk.append(line)
print("".join(chunk))  # or whatever you want to do with it

.

>TRINITY_DN63782_c0_g1_i1 len=433 path=[411:0-432] [-1, 411, -2]
ATAGACACGAACACAAACACATAAATAATTTGAGAAAATAGAAGTGATTGAACTTGTTGG
TGTGGTACAGGTGTCAAACAAACCTTCAACCAGAAGTTTTGTTGCTGCATAAATCATAGT
GACACTCTGATATGATATCAAAGAAAATCATGTAACCCAAATACATCCCTAAGTATCTAG
TTGAAGCTACAGTCCACTAATTGTAACAATATTAAGTAATTATGAAATGAACCATTTGCA