The word abide is closely connected to the words house and home in scripture (Luke 19:5).  The invitation to “abide with Me” is an invitation to be “at home with Me”.  With that in mind, I offer the following paraphrase of the Vine and the branches.  “Make your home in Me, and I will make my home in you.  As the branch cannot bear fruit of itself unless it is at home in the vine, so neither can you unless you are at home in Me. I am the vine, you are the branches; he who is at home in Me and I at home in him, he bears much fruit, for apart from Me you can do nothing” John 15:4-5.  Jesus is the Home of the Soul and by incarnation has chosen to make us His home.  Wow!  Oh the beauty of this mutual abiding!