When does the second polar body gets expelled from the egg nucleus during oogenesis ? I know that it occurs after the entry of the sperm into the secondary oocyte but does it occur before the fertilization or in the period between the entry of the sperm and fertilization ?
As stated in the answer by @CDB, these polar bodies don't last too long, and the second polar body which forms as the result of meiosis II, extrude only after fertilisation. The secondary oocyte is arrested in metaphase II until it becomes fertilised, so meiosis II can only by completed after ovulation and fertilisation (see e.g. UNSW Embryology: Meiosis at https://embryology.med.unsw.edu.au). This means that meiosis II, which actually forms the second polar body, occurs as a result of fertilisation. The secondary oocyte becomes a mature oocyte (ovum) by the expulsion of second polar body into perivitelline space.
The metaphase arrest is caused by the presence of two types of cytoplasmic activity 1. MPF - maturation promoting factor 2. CSF - Cytostatic factor that inhibits APC/C
Only when MPF is present, stabilized by CSF, and active will the M phase persist. However, this complex is very sensitive to calcium, such that a rise in calcium tilts the equilibrium away from complex synthesis towards destruction.
Capcitated spermatozoa that have undergone an acrosome reaction is capable of fusion and Calcium mediated calmodulin is required for this process. When fusion happens spermatozoon ceases to move and the nuclear with parts of mid piece and tail passes to the ooplasm. So sperm initiate Calcium release at fertilisation this helps in the completion of the 2nd meiotic division. For further detail on these processes you could read the article Penetration, Adhesion, and Fusion in Mammalian Sperm-Egg Interaction Increased Calcium activated Calpain (calcium dependent cysteine protease which causes CSF to disappear thereby activating APC/C which in turn causes dissociation of MPF. This results in completion of 2nd meiotic division.
For further details I recommend you to read Johnson and Everitts Essential Reproduction (6th edition), cell cycle chapter in The cell by Bruce Albert (5th edition)
During oocyte–sperm fusion and second polar body expulsion, the cytoplasmic contents of the sperm cell membrane (now fused with the oocyte membrane) pass into the oocyte cytoplasm. Between 4 and 7h after fusion, the two sets of haploid chromosomes each become surrounded by distinct mem- branes and are now known as pronuclei (see Fig. d). The male is usually the larger of the two. Both pronuclei contain several nucleoli. During the next few hours, each pronucleus gradually moves from its subcortical position to a more central and adjacent cytoplasmic position. During this period, the haploid chromosomes synthesize DNA in preparation for the first mitotic division, which occursabout 18–24h after gamete fusion. The pronuclear membranes around the reduplicated sets of parental chromo- somes break down (see Fig. e), the mitotic metaphase spindle forms and the chromosomes assume their positions at its equator. The final phase of fertilization has been achieved: syngamy (or coming together of the gametic chromosomes) has occurred. Immediately the first mitotic anaphase and telophase are completed, the cleavage furrow forms, and the one-cell zygote becomes a two-cell conceptus.
A secondary oocyte contains n/2C genome and a sperm fuses with a cell with n/C genome (matured ovum). Therefore a secondary oocyte must divide and eject a polar body and that happens due to fertilisation. so the answer is after fertilisation and before entry of sperm.
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Polar bodies will usually undergo apoptosis in about 17 to 24 hours after the egg forms (the final stages of telophase II (cytokinesis)) because they have relatively little cytoplasm and shunted organelle development (Schmerler & Wessel, 2011).