Parliamentarian Tynchbek Akmatbaev had gone to inspect the Moldovanovka prison, when inmates -- many of whom have tuberculosis -- began rioting to demand better living conditions. Akmatbaev was shot dead on 20 October under circumstances that remain unclear.
Today, protesters in Bishkek have demanded Kulov's resignation, blaming him for Akmatbaev's death.
The rally drew hundreds of protesters to the capital's main square. Placards carried a clear message for the country's prime minister, demanding "Kulov resign" and charging "Kulov is a killer."
Among those demonstrating was the slain deputy's brother, Rysbek Akmatbaev. He claimed that Kulov had tried putting the blame for the prison incident on the victims.
Kulov Issues Warning
But Kulov countered by rejecting the protesters' allegations. He said he would only step down if the president and the parliament dismissed him, as required by law.
"We will do everything according to the law. If parliament wants to look into the protesters' demands, please let it do so," Kulov said. "I'm even asking parliament to look into the protesters' demands. I'm also asking the president to look into the protesters' demands. Their demands should be examined according to the law. If parliament, or the president, finds grounds for my resignation, then there will be no objections from my side. But I want to stress again, we must do everything according to the law. This is what I'm asking for."
And Kulov warned that authorities would act swiftly if protesters tried to stir up trouble.
"Everything will remain within the framework of the law," said Kulov, who himself swept to power along with the former opposition after public unrest that culminated in the ouster of longtime President Askar Akaev. "But I want to stress that if this is an unlawful gathering, if it will destabilize the situation, adequate measures will be taken [on the part of authorities]."
Akmatbaev is the third member of parliament to have been killed in Kyrgyzstan since June.
The violent incident put the spotlight on conditions in Kyrgyzstan's prisons.
Parliament voted on 21 October to quickly allocate extra money to make improvements.
Kulov said today the government is working on the material needs of the prisons. There are, he said, a lot of financial problems in Kyrgyz prisons.
Today's events suggest the prison incident might be turning into a political problem, too.
By evening, authorities had persuaded the anti-Kulov protesters to move from Bishkek's main square. But they initially moved only a short distance away, in front of parliament.
Some opposition lawmakers have demanded an extraordinary parliament session on 23 October. If they succeed, they say they will use the event to call for Kulov's ouster.
(with additional news agency reporting)