In London, a new play about President Vladimir V. Putin of Russia and a revival of “The Seagull” explore undercurrents of pain.LONDON — There’s a chill in the air at the Almeida Theater, notwithstanding the record-breaking heat here. That drop in temperature comes from the coolly unnerving “Patriots,” a new drama whose look at power politics in Russia over the last quarter-century induces a shiver at despotism’s rise.The gripping production, directed by Rupert Goold, runs through Aug. 20.Written by Peter Morgan (“The Crown,” “Frost/Nixon”), “Patriots” surveys the sad, shortened life of Boris Berezovsky, the brainiac billionaire who died in 2013, age 67, in political exile in London. An inquest into Berezovsky’s mysterious death returned an unusual “open verdict,” but on this occasion, it is unequivocally presented as a suicide: The play ends with this balding man, bereft of authority, preparing to end his life.An academic whiz-turned-oligarch who expedited the rise of the younger Vladimir V. Putin, Berezovsky later fell out with the onetime ally who enlarged his power base, according to the play, with promises of “liberalizing Russia,” yet proceeded to do anything but.Morgan introduces Berezovsky, age 9, as a math prodigy whose mother hoped he might become a doctor. (A gleaming-eyed Tom Hollander plays the role throughout.) From there, we move forward 40 years to find Berezovsky an integral member of Russia’s moneyed elite welcoming to his office an obsequious Putin, then deputy mayor of St. Petersburg.“Respected Mr. Berezovsky,” says an initially indrawn, ferret-like Putin, “one would have to live on another planet not to know you!” But it isn’t long before Putin has changed his tune, and his tone, as he rises from prime minister to president and consolidates power around himself. In one notably effective wordless scene, Putin tries out poses in front of a mirror to see which makes him look most impressive. His earlier hesitancy has given way to a man in love with his own heroism.Berezovsky looks on at so dramatic a change in character appalled, urging the former K.G.B. operative to “know your place.” But Putin by this point simply won’t be sidelined. And besides, reasons Putin, why hold your enemies close when they can just as easily be destroyed?Tom Hollander as Boris Berezovsky in “Patriots.”Marc BrennerGoold, the director, dealt with a different headline-maker at the Old Vic this spring in “The 47th,” which imagined Donald J. Trump in the run-up to the next presidential election. Goold is in better company this time: “Patriots” is a richer, less fanciful play, with grim resonances for today. Although Morgan rightly leaves it to the audience to make the connection, you can draw a line between the glorious empire Putin yearns for in the play and his ongoing attack on Ukraine.In one of the performances of the year, Will Keen, as the Russian leader, astonishes throughout, bringing his character to agitated, unpredictable life. His early fawning in Berezovsky’s presence gives way to an icy rejection that finds its fullest expression when his onetime mentor writes as a fellow patriot requesting permission to come home to Russia. Putin dictates a reply, then tells his secretary to rip the letter up: Berezovsky, Putin concludes, “is not worth it.”Hollander impresses, too, as he did in a dazzling star turn in “Travesties,” which won the actor a 2018 Tony nomination — two talky plays requiring an actor at home with reams of language. His character is both a quick-tempered womanizer, and too naïve to realize the young Putin’s potential for authoritarian misrule.Widening the play’s scope yet further is the Russian president’s friend, the oligarch Roman Abramovich (the excellent Luke Thallon), who battles Berezovsky over ownership of the oil company Sibneft. That case, which came to trial in London in 2012, plays out here as a resounding defeat for Berezovsky that only amplifies his psychic distress. Alexander Litvinenko (Jamael Westman, a former leading man in “Hamilton”), the Putin critic who was poisoned in 2006, shows up, too, as the “most honorable” of dissidents (or so Morgan maintains): a political casualty wreathed in glory that the sorrowful Berezovsky never knew.There’s an aspect of bravery, you feel, in writing “Patriots” at all while Putin is on the march. (That said, like Trump with “The 47th,” it’s possible these men’s egos would thrive on the attention.) In the days after Russia launched its attack on Ukraine, orchestras, concert halls and opera houses pulled Russian works from their stages, and it looked as if it might no longer be allowable to perform the Russian repertory in the West; overseas trips by the Bolshoi Ballet, among other storied Russian arts companies, were canceled, as well.Emilia Clarke, second from right, in Anya Reiss’s interpretation of Anton Chekhov’s “The Seagull,” directed by Jamie Lloyd at the Harold Pinter Theater.Marc BrennerSo it’s a relief to welcome a Russian classic, “The Seagull,” first presented in 1896 by Anton Chekhov, who died nearly a half-century before Putin was even born. That this first of Chekhov’s four great plays ends, as does “Patriots,” with a suicide is an intriguing coincidence that also points to the undercurrents of pain that inform both plays.Performed barefoot and in modern dress, Jamie Lloyd’s enthralling production, at the Harold Pinter Theater through Sept. 10, furthers the stripped-back approach to the classics he brought to a recent “Cyrano de Bergerac” that was acclaimed in New York and London.Just as that play dispensed with a fake nose for its title character, this “Seagull,” seen here in Anya Reiss’s 2012 version, never features the wounded bird of the title onstage. Doing without props of any kind, the cast members, headed by the “Game of Thrones” alumna Emilia Clarke in a terrific West End debut, deliver the play seated on green plastic chairs and boxed in by chipboard; they speak with a quiet intensity, as though we were eavesdropping on the characters’ innermost thoughts. Some will be exasperated by the approach, but I was riveted from the first hushed utterance to the last.Like “Patriots,” this “Seagull” draws from its own well of grief, even if the world of writers and actresses in Chekhov’s play is a long way from Morgan’s power-brokers and politicos. Lloyd’s ensemble communicates the shifting affections of a quietly devastating play that leaves you transfixed by the theatrical potency of despair.Patriots. Directed by Rupert Goold. Almeida Theater, through Aug. 20.The Seagull. Directed by Jamie Lloyd. Harold Pinter Theater, through Sept. 10; in cinemas Nov. 3. More