Westminster attacker Khalid Masood acted alone and there is no information to suggest further attacks are planned, Scotland Yard has said.
Masood killed three people and injured 50 when he drove a car into pedestrians on Wednesday. He then fatally stabbed a police officer before being shot dead by police - all within 82 seconds.
Police said it was possible "we will never understand why he did this".
The family of PC Keith Palmer said his bravery would be remembered.
MP Tobias Ellwood, who was among those who tried to save PC Palmer's life, has said he is "heartbroken" that his efforts were not successful.
Deputy Assistant Commissioner Neil Basu said: "We still believe that Masood acted alone on the day and there is no information or intelligence to suggest there are further attacks planned.
"Even if he acted alone in the preparation, we need to establish with absolute clarity why he did these unspeakable acts to bring reassurance to Londoners, and to provide answers and closure for the families of those killed and the victims and survivors of this atrocity.
"We must all accept that there is a possibility we will never understand why he did this. That understanding may have died with him.
"Nevertheless, we are determined to understand if Masood was a lone actor inspired by terrorist propaganda or if others have encouraged, supported or directed him."
Writing in two Sunday newspapers, both the home secretary Amber Rudd and Foreign Secretary Boris Johnson called on social media companies like Google and Facebook to be "more proactive" in stamping out extremist websites.
In a statement released on Saturday, the family of PC Palmer addressed those who tried to save his life.
"There was nothing more you could have done," it said.
"You did your best and we are just grateful he was not alone.
"We care about him being remembered for his selfless bravery and loving nature. We miss him so much, but we are also incredibly proud of Keith."
The family said they had been "overwhelmed by the love and support" shown for them and for PC Palmer and they praised the support from the police.
- 14:40:08 - the car that Masood was driving over Westminster Bridge first mounted the pavement on the northbound side
- 14:40:38 - after continuing towards Bridge Street along both the footpath and road, Masood crashes into the perimeter fence of the Palace of Westminster
- 14:40:59 - the first 999 call was made to the Met police reporting the incident
- 14:41:30 - Masood left the vehicle and was shot by a police firearms officer inside the Palace of Westminster
In a statement, also released on Saturday, Mr Ellwood said: "I am heartbroken that I could not do more for PC Keith Palmer who gave his life in holding the line against terrorism and defending democracy."
The Foreign Office minister said he was "deeply humbled and overwhelmed" by messages of support.
"I played only a small part that day, doing what I was taught to do, and am honoured to have been invited to join the Privy Council afterwards." he said.
"It is right that we concentrate our thoughts on the victims as we stand side by side to protect all that we hold dear, including our precious values and way of life which will always prevail."
Masood's other victims were Aysha Frade, who was in her 40s and worked at a London sixth-form college, US tourist Kurt Cochran, 54, and retired window cleaner Leslie Rhodes, 75, from south London.
A 58-year-old man, who was arrested in Birmingham the morning after the attack under the Terrorism Act, remains in custody, and a 32-year-old woman who was arrested in Manchester, remains on police bail, Metropolitan Police have said.
Eleven people were initially arrested over the incident and nine people in total have been released without charge.
The Metropolitan Police said Masood, 52, who had previous criminal convictions but none for terrorism, had used a number of aliases.
At birth, he was registered in Dartford, Kent, as Adrian Elms, but later took his stepfather's name becoming Adrian Ajao in childhood.
In the early 2000s, he was convicted of causing grievous bodily harm after slashing a man across the face with a knife in a pub.