MK One is the latest in a stream of retailers that have required intervention from their private equity backers.
Baugur has parachuted in Kevin Lyon, former head of venture capital giant 3i's Scottish division, to chair the retailer. He will run the chain until a replacement is found.
Baugur acquired the 170-store chain, formerly called Mark One, for£55 million including debt in November 2004. Iceland's second-largest bank, Landsbanki, took a 36 per cent stake in the group at the time.
Johnston, who became chief executive at the time of the deal, had previously held roles at Dunnes Stores, Burton Group and Debenhams.
MK One has faced stiff competition from the value end of the market, with ABF-owned Primark piling on the pressure. A source close to MK One said: 'The company hasn't done as well as it should. It's been a tough period since Baugur bought the business.'
Despite problems, a trial fascia rebranding to MK is thought to be performing well. Eight locations, including Woking and Nuneaton, have been rebranded in an attempt to reposition the business as a young fashion retailer, rather than a discount chain.
The branding change followed the appointment of MK One's first buying director, Suet Cheung, who Johnston poached from Miss Selfridge last year to drive its fashion ranges.
Baugur's troubles with MK One have not deterred its continued hunt for further retail acquisitions. On Monday, it got the green light from shareholders for a£311 million bid for garden centre group Wyevale in a consortium with Scottish retail entrepreneur Sir Tom Hunter. It is also widely thought to be preparing a possible offer for House of Fraser.
Retail has not been a bed of roses for private equity firms of late. Value clothing chain Ethel Austin and footwear specialist Faith both required refinancing packages from VC backers earlier this year and Apax-owned entertainment retailer Silverscreen went into administration in March.
No comment was available from Baugur.