if i have understood well, you are hydrolyzing lysine (i guess by an enzymatic process) and the you want to recover the fraction corresponding to such protein and valorize its activity.

Well if you are doing that I am afraid that you are only recovering peptides, hence you will have no activity, since you has broken down the lysozyme into smaller peptides with no antimicrobial activity (at least not enzymatic activity as it was in the whole lysozyme).

If you wan to recover such peptides you can follow different strategies depending on the raw material:

if it was pure lysozyme you can use solid phase extraction, ultrafiltration, ionic exchange chromatography, electrophoresis, etc.

if it was a mixture of proteins the thing is more complicated and you will need to combine different purification techniques coupled to mass spec identification.

I hope it helps a little.

i know that after  hydrolysis will loss the enzymatic activity of lysozyme but i make hydrolysis to obtain  the peptide which responsible for its activity depending on (Ibrahim et al 2001) i attached the link  for you .

http://www.ncbi.nlm.nih.gov/pubmed/11560930

but the main problem for me how can i deal with the fractions after its separation by chromatography to determine its antimicrobial activity.

After chromatogrpahy separation, if you have identified the peak corresponding to the peptides which posses the activiy, i would pool several of these peaks to have a high amount of the peptie of interest. After that, maybe you need to remove salts from the buffer, to do that you can use SPE (HLB or C18 cartridges). Once you removed the salt you can evaporate the solvent (usually acetonitrinile or methanol) under nitrogen stream and moderate temperatures, and you will have a purified dry peptide. SUch peptide will be ready to be used in the antimicrobial activity.

what means by SPE?

Solid phase extraction