When a new publicly traded company's stock price takes off like a rocket – as Twitter's (NYSE:TWTR) did on its first day of trading, shooting up to over $70 a share – investors' expectations also spike. So it's no wonder that Twitter shareholders, after the wild ride of the past year in which its stock has been as low as $29.51 a share and as high as $71.25, are doing more than just grumbling -- many are calling for (more) heads to roll.

Several Twitter executives have already gone to the chopping block as CEO Dick Costolo struggles to find the right combination of leadership and direction. This begs the question: has the time come for a new CEO, someone who can formulate and implement a comprehensive plan to address Twitter's many issues? Many investors and industry pundits seem to think so, and they might be right.

The pressure is mounting
Around the time Twitter introduced finance all-star Anthony Noto as its new CFO over the summer, the whispers regarding Costolo's future were already growing louder. While Noto was widely viewed as a sound addition to management, Twitter's rapid executive changes raised more questions than they answered.

Noto's hiring came shortly after several key folks were either ushered out the door or left of their own volition. Some of the more notable departures included operations chief Ali Rowghani, who was followed by his most senior manager on the day of his abrupt departure. Add a new engineering chief and product exec to Twitter's management shuffle, and it's no wonder investors began wondering what was happening

Unfortunately for Costolo fans, it gets worse. Some big-time institutional investors have become furious, to put it mildly, with the CEO's continuous selling of his Twitter stock. Just months after saying he would not sell his stock, Costolo started unloading his personal stash -- shedding 500,000 shares of Twitter from his family trust's holdings.

With its stock price battered, wouldn't a CEO bent on reinvigorating his company and enhancing shareholder value be on buying spree? He would if he believed a turnaround was inevitable. Costolo responded that he still holds a huge position in Twitter, which is true, but there's no denying the message sent by his recent stock sales. As one Twitter shareholder put it, "As the CEO, how do you look the employees in the eye when you are busy grabbing a lifeboat?"

The defining moment
If it turns out Costolo is heading for the exit, as some predict, we might look back at the recent rant by co-founder and Twitter board member Evan Williams as the moment when the wheels finally came off. Why? Because among Williams' complaints, primarily regarding comparisons with Facebook (NASDAQ:FB) and its fast-growing Instagram property, was his displeasure with investors' and industry pundits' reliance on monthly active users, or MAUs, as a metric for growth in a social-media company.

As noted in a recent article, the problem with Williams dismissing Twitter's MAU concerns, especially as they relate to Facebook and Instagram, are twofold: one, it comes off sounding like a kid on the playground screaming "it's just not fair!"; second, and more importantly, Williams' statements contradicted remarks made weeks earlier by Costolo during Twitter's first-ever analyst day. The CEO spent much of the nine or so minutes of his presentation discussing plans for boosting MAUs, improving the user engagement experience, and exploring video and other options to accomplish both.

These very public statements, and their glaring differences create the perception that the left hand of Twitter's management team does not know or care what the right hand is doing. Add in wholesale management changes and Costolo shedding of hundreds of thousands of shares and the argument that now is the time for change becomes pretty strong.

Need further proof that investors have had enough of Costolo? Twitter's share price jumped 4% the day an analyst suggested there's a good chance "he's [Costolo] not there in a year." Right or wrong, it's time for a change if Twitter's stock price is going to live up to admittedly high expectations anytime soon, and that change needs to start at the top.