John's favorite, and only book, as a child was The Complete Works of William Shakespeare, so it is appropriate that he would use it as a basis for his comparisons. When John first meets Lenina he is entranced by her. He view their differences as the result of their being raised in different places by different people. However, he is still optimistic about their relationship despite the odds stacked against them.
Later, John becomes disillusioned with Lenina and is filled with disgust for her. He calles her the same names that Othello called Desdemona, and though he does not kill her, he is finished with her forever.
And as though awakened by her cry he caught her by the shoulders and shook her. ‘Whore!’ he shouted. ‘Whore! Impudent strumpet!’
The relationship moves, for John, from one of hope and anticipation to one of disillusion and realization just like the tragics couples from Shakespeare.