Image caption Some have seen this as a farewell visit by Mrs Clinton to Israel

In Cairo and Jerusalem, over three days of meetings and talks, US Secretary of State Hillary Clinton appealed for patience as one country continues its bumpy transition to democracy and its neighbour frets about the impact this has on the relationship between them.

Israel is very anxious about the continued upheaval in the region and worried about the rise of Islamists to power, especially in neighbouring Egypt, which has just elected an Islamist President, Mohammed Morsi.

Egypt has a peace treaty with Israel and Israel's leadership is worried about the fraying of the agreement over time.

Israeli President Shimon Peres and Prime Minister Benjamin Netanyahu have reached out to the new Egyptian leader, sending him congratulatory letters and pressing on him the importance of maintaining the Israel-Egypt peace treaty signed in 1978.

Mr Morsi has yet to respond to their overtures, and Mr Peres and Mr Netanyahu have had to satisfy themselves with reassurance by a messenger.

Mrs Clinton told the BBC in an interview that she had shared with Israeli leaders that Mr Morsi had told her in private what he has been saying in public- that he would abide by the peace treaty with Israel. But she added that what mattered now was action, not words.

Mr Morsi has also said that Egypt will abide by agreements it is party to, as long as the other side upholds their part of the treaty.

"The amount of work ahead of this Egyptian government would be daunting for the most experienced leaders Hillary Clinton

The Camp David accords make specific reference to comprehensive peace between the Israelis and the Palestinians - this gives Egypt wiggle room over time.

Farewell visit

As anxious as Washington itself is about ensuring that the peace treaty continues to be respected, Mrs Clinton appeared to counsel Israelis to be patient and understand that reassuring Israel was not necessarily the first item on Mr Morsi's agenda.

"The amount of work ahead of this Egyptian government would be daunting for the most experienced leaders," she said during a news conference in Jerusalem at the end of a long day of talks with Israel's top leadership.

This is being seen as a farewell visit to Israel as the end of her tenure approaches.

"The economy is in desperate need of reform, the political process is a work in progress, a long way from being finalised, there are serious fissures within society that have to be addressed," she said, pointing out that Egypt was going through this democratic process for the first time in 5,000 years.

Israel is trying to come to terms with the changes sweeping the region, and the dire public warnings by Israel during the Egyptian revolution about the dangers of an Islamic takeover if Hosni Mubarak fell have been replaced with a more subdued wariness.

Image caption Ms Clinton's meeting with Mr Mursi is part of the US's cautious engagement with Islamist politicians

Israeli officials now also point out that Hosni Mubarak maintained a very cold peace with Israel and only visited once during his 30-year rule - for the funeral of Yitzhak Rabin.

Cautious engagement

Washington is also feeling its way around the new Egypt and the new Arab world.

During the Bush administration, then Secretary of State Condoleezza Rice said the US had not and would not engage with the Muslim Brotherhood.

After encouraging elections in the Palestinian territories in 2006, the Bush administration rejected the victory of Hamas, the Palestinian militant group.

The Obama administration has had to accept the inevitable in Egypt, and the results of the presidential election, especially after constantly stressing that the future of Egypt was in the hands of the Egyptians.

However, this was only possible because the Muslim Brotherhood has renounced violence. The US is now engaging cautiously with Mr Morsi, hoping this will influence some of the choices he makes.

Mrs Clinton repeatedly called for respect for minorities and emphasised that democracy was not just one election, but a process. She reminded Egyptians that the peace treaty with Israel was good for their country because it had allowed a generation of Egyptians to grow up without war.

Mubarak could be a stubborn US ally, often refusing American advice or exhortations and doing little to soothe anti-Western feelings in his own country.

But in the post-Mubarak Egypt, US influence is even more limited, with both the army and the Islamists pushing back against Washington as a way of asserting their own power. But both are still keen on American aid.

Mrs Clinton also appealed for patience in Cairo, telling Egyptians their democratic transition was not unique and that countries from Latin America to Asia have navigated this bumpy path and worked their way out of military rule.

Noisy protests

Mrs Clinton seemed at turns frustrated and surprised by all the conspiracy theories making the rounds on the streets and in the Egyptian media.

She spent a lot of her energy denying that the US had picked a side and pointed out repeatedly she was meeting all stakeholders in the country - the president, the army and civil society.

Image caption Protesters have accused Hillary Clinton of favouring the Muslim Brotherhood

Egyptians who either fear the Muslim Brotherhood, like Christian Copts, or feel they are on the losing side in the new Egypt, like Mubarak loyalists, are convinced the US has struck a deal with the Islamist group.

Outside her hotel in Cairo, demonstrators held up placards with slogans saying: "We don't want Hamas to rule Egypt", a reference to the conservative rule the group - which was inspired by the Muslim Brotherhood - has maintained over Gaza.

Another slogan, "Obama don't send your dollars to the jihadis" seemed to indicate a fear that the whole country, including the army, was now in the firm grip of Islamists.

In Alexandria, Mrs Clinton's motorcade was pelted with tomatoes and shoes. No-one was harmed and Mrs Clinton said she was not offended by the protests or their slogans, pointing out that protesters were not a uniquely Egyptian phenomenon.

She said that seeing people express themselves, even if their assumptions and conclusions are wrong, is a sign that democracy is thriving.

But she added it was clear that Egyptians were uncertain about the path forward.