What part did Mary’s biology play in the making of the person of Jesus Christ?

Since Jesus has both divine and human natures the church has taught that Jesus is both “Son of God” and “Son of Mary.”  Your question as to how this could be was one that also puzzled Mary as she asks “How can this be since I do not know man?” (Luke 1:34)
The angel’s response tells us a bit of what an answer to your question could be:  “The holy Spirit will come upon you, and the power of the Most High will overshadow you. Therefore the child to be born will be called holy, the Son of God.” (Lk 1:35)
However, you have asked a more biological question in which the Catechism of the Catholic Church helps to clarify:

485 The mission of the Holy Spirit is always conjoined and ordered to that of the Son. The Holy Spirit, “the Lord, the giver of Life”, is sent to sanctify the womb of the Virgin Mary and divinely fecundate it, causing her to conceive the eternal Son of the Father in a humanity drawn from her own.” (CCC, 485)
The word “fecundate” means to impregnate.  So the best explanation we have lies here that is was the power of the Holy Spirit that makes her pregnant.
That implanting of a divine nature with Mary’s female human egg that starts Mary on the cycle of pregnancy.
I hesitate to say that it was “divine sperm cells” that unite with the egg because that is how we might understand this scientifically in our human experience and indeed this may be how the Holy Spirit was able to impregnate Mary–but we don’t really know that for certain.
This is a miracle, something that confounds science and even goes beyond it into something of a mystery.  This is divinity and humanity in union–which goes beyond our human experience. In short, God enables new life to start by uniting in some way with Mary’s humanity.
Suffice it to say that the power of the Holy Spirit makes Mary pregnant.  There isn’t a partially grown fetus implanted but rather Jesus shares in our human experience in all its wonder from the union of two cells which begin to divide, through birth, into life and into our death and then beyond it.