Father told of deaths of his son and daughter
Their parents, Thomas, 55, and Angela, 46, suffered serious burns and are under sedation in hospital.
Yesterday police said Mr Sharkey, who is in Glasgow Royal Infirmary, has now been told of his children's deaths, but that his wife, who has been moved to the same hospital, does not yet know. Both were said to be in a critical condition.
Strathclyde Police said more than 50 officers are working on the murder investigation, while Crimestoppers is offering a 10,000 reward for information leading to the arrest and conviction of whoever committed the crime.
Detective Chief Inspector Anne McKercher, of Strathclyde Police, said: "We have a dedicated team of officers currently working on this investigation, following a number of lines of inquiry.
"Officers are continuing to speak to local people within the community, conducting door-to door inquiries and gathering any information which could help us trace whoever is responsible.
"A family has been left devastated and they are doing their best to try and come to terms with what has happened. We will continue to pursue all areas and I would still like to continue my appeal to people to contact us.
"Any piece of information, no matter how insignificant it may appear, could prove vital in our investigation."
Kate Jackson, national manager for Scotland for the Crimestoppers Charity, said that the response to the appeal for information had been "encouraging".
"We are hoping the reward will encourage someone to come forward with vital information to solve this horrible crime," she said.
"It is a dreadful tragedy and this level of reward of up to 10,000 is based on the circumstances involved in what is a truly a horrific crime. The response from the public has been very encouraging and it shows the horror that the community feels about what has happened. Quite clearly, people feel very strongly about it and they want to do something to catch those responsible."
Emergency services were called to the fire at the flat in Scott Court at about 5:10am on Sunday. The fire broke out in the living room and quickly spread.
Police said the building had just one door and that the family had become trapped by the fire.
Mr Sharkey senior was seen hanging out of the window and shouting for help. His son was found dead at the scene. The rest of the family was taken by ambulance to Royal Alexandra Hospital in Paisley, Renfrewshire, where Bridget died a short time later.
She was a pupil at Helensburgh school St Joseph's Primary, whose headteacher Ann Chalmers described her as "a beautiful child, very talented, an excellent all-rounder and enthusiastic pupil".
Thomas Sharkey, below left, won a golf scholarship to Georgia Southern University in the US, after leaving Our Lady and St Patrick's High in Dumbarton in 2008.
In his biography on the website of Georgia Southern Eagles, the university's athletics group, he described himself as "ambitious and competitive", adding that his grandfather and father were the biggest influence on his career for teaching him golf.