How are family dynamics/interactions integral to the plot of A Tale of Two Cities by Charles Dickens?

In his highly structured A Tale of Two Cities, Charles Dickens employs the motif of pairings and counterparts throughout his narrative. Certainly, the interactions among the families of the Manettes and the Evrémondes and the family of Madame deFarge are central pairings in the novel. 
The intricate workings of the family interaction begin with the voyage to Calais, France by Lucie Manette and the venerable Mr. Lorry of Tellson and Company, who acts as her adviser and chaperone. In Paris, Lucie meets her father for the first time since he has been hidden away as a prisoner in the Bastille for eighteen years, having been abducted and placed there because of the dastardly Evrémonde twins. He is "recalled to life," as it were, and is cared for by Ernest Defarge, who was Manette's servant long ago. After Lucie Manette arrives in St. Antoine at the Defarges' wine shop and meets her father, she takes him back to England with her so that he can recuperate from the nearly two decades of deprivation that he has suffered.
Manette's tragic past is also connected to the vengeful and cruel Evrémondes, aristocrats whose pasts are additionally tied to the family of Madame Defarge, a family who were peasants on their estates, and who died because of the cruelty of these Evrémonde twins. For, it was Madame Defarge's sister who was ravaged and killed by these arrogant and cruel men, and her brother who died trying to defend this sister. Shortly thereafter, the father died from grief.
Ironically, the history of the Evrémondes is also resurrected after Lucie meets Charles Darnay during her passage back to England, as well as in the court at the Old Bailey after Darnay is accused of treason. From this meeting of Charles Darnay and Lucie Manette there develops a love relationship. When Darnay comes to Dr. Manette to ask for the hand of Lucie in marriage, he tells the old physician that he has a secret to reveal about himself. But, Manette refuses to listen to his revelation until after he and his daughter are married. Therefore, on the wedding day, Charles Darnay reveals what Dr. Manette has feared. That is, Darnay's real name is Evrémonde; he is the son of the twin who has died. Also, Charles informs the doctor that he has renounced his family name because of his father and uncle's nefarious actions.
If the Manettes and the Darnays were all to have remained in England, the past may have faded somewhat from their memories and lives. But, Charles is summoned back to France after he receives a letter from his overseer, who has been arrested by the revolutionaries and implores Darnay's testimony that he no longer taxes the peasants on his land. Unrealistically, too, Darnay hopes to influence the radicals to more reasonable actions. Of course, once the revolutionaries learn that he is Charles Evrémonde, they arrest him. When Dr. Manette, revered by the new government as the Bastille prisoner, tries to save Charles from the guillotine, an earlier document that was written while he was a prisoner in the Bastille surfaces. In this, Manette condemned the Evrémondes, declaring that the entire family should be "exterminated." So, Dr. Manette unfortunately harms, rather than helps, Charles, who is then taken to prison.
When he learns of Darnay's fate, Monsieur Defarge implores his wife to have some sympathy for this innocent son of the cruel Marquis d' Evrémonde, who has renounced his name and his family, but she wrathfully tells her husband, "Tell the Wind and the Fire where to stop; not me!" (Bk. 3, Ch. 12). Later, it is this misfortune of Charles Darnay that allows the dissipated Sidney Carton to intervene and keep his promise to the woman he adores, Lucie Manette. Having promised Lucie that he would "do anything" for her and her loved ones, Carton, who resembles Darnay, manipulates one of the spies at the prison and switches places with Darnay by drugging him and having him carried out. The Manettes then leave France and return to England as Madame DeFarge, too, fails in her attempt to kill Lucie and avenge herself.
Clearly, the interweavings of family interactions are intrinsic to the plot and essential to the development of themes in A Tale of Two Cities.