House Democrats officially gained two and lost two on Thursday as they continued their painstaking zigzag toward 216 votes and final passage of a sweeping health care overhaul, now likely on Sunday.

Leaders got two pieces of good news, with retiring Science and Technology Chairman Bart Gordon (D-Tenn.) and freshman Rep. Betsy Markey (D-Colo.), who voted “no” on the original House bill, announcing they would back reform this time around. But those gains were offset by the losses of Reps. Michael Arcuri (D-N.Y.) and Stephen Lynch (D-Mass.) who flipped their previous support to opposition.

Lynch, in particular, was a blow to reform supporters, since the former union president had not been on any watch lists of potential vote flippers. He ripped the Senate-passed measure as a “surrender” to insurance companies, and he held fast to his opposition even after a meeting at the White House with President Barack Obama on Thursday afternoon.

Though Obama failed to sway Lynch, House Democrats were buoyed by the whip in chief’s decision on Thursday to postpone his planned Asia jaunt until later this year so he can be on hand to help round up support for his signature domestic initiative.

The release of the final bill text — and estimates from the Congressional Budget Office — gave handfuls of other undecided lawmakers some homework to do, as many on the fence continued to reserve judgment until reading through the language and scoring. But the CBO’s deficit reduction projections — a whopping $1.2 trillion in the second decade — had some former opponents encouraged. “The numbers sound promising, but I want to see how we got to the numbers,” said Rep. Brian Baird (D-Wash.), who opposed the House-passed bill.

The arithmetic for Democratic leaders is simple: If they can hold on to all 216 of the Democrats who voted for House passage and are still serving, they win. But rounding up those supporters has proved tricky. And they need to offset every defection by converting someone who voted against the original measure.

Rules Chairwoman Louise Slaughter (D-N.Y.) dismissed the threat of minimal defections. “We’ve got a surplus. We’ve got some spares,” she said. Speaker Nancy Pelosi (D-Calif.) likewise projected confidence, telling reporters at a midday press conference, “We feel very strong about where we are.”

Others were measured. Energy and Commerce Chairman Emeritus John Dingell (D-Mich.) said support for the legislation “looks good and feels good.” But the veteran health care reform advocate cautioned that this was “the murkiest time” in the vote-counting process.

The gravest threat remains abortion language in the Senate version that some anti-abortion rights Democrats argue doesn’t go far enough to block taxpayer dollars from funding the procedure. Rep. Bart Stupak (D-Mich.) has led opposition to the provision, but just how many lawmakers are behind him remains unclear, and leaders are hoping they can hold abortion-related defections down to about five.

Rep. Marcy Kaptur (D-Ohio), a Stupak supporter though officially still undecided on the Senate bill, on Thursday urged Pelosi to hold a separate vote reaffirming the stricter House approach to the matter. “I want to be constructive,” she said. “We have to find a way to work it out.” But abortion rights supporters rejected the proposal out of hand. “We’re not going to do that,” said Rep. Diana DeGette (D-Colo.), co-chairwoman of the Pro-Choice Caucus. “We don’t know it would die in the Senate. The stakes are too high.”

The threat of another social debate derailing reform fizzled on Thursday when Rep. Luis Gutierrez (D-Ill.), who was officially opposed to the latest incarnation of the overhaul on account of tougher immigration language, announced his support. The Chicago Democrat had signaled as much on Wednesday, when he said he was taking a “macro” approach to his vote and suggested he could come around if Obama gave him assurances that the administration would push for comprehensive immigration reform in short order.

His announcement came as the full Congressional Hispanic Caucus announced its endorsement of the health care bill.

“I cannot see that voting against this health care bill is going to bring us any closer to comprehensive immigration reform,” Gutierrez said. “I do see that a success and a victory on health care will allow this president to be strengthened and to be able to carry out with more political capital our ultimate goal.”

Leaders scrambled, meanwhile, to defend their preferred procedural strategy for passing the bill. The “Slaughter Solution” — by which Democrats would avoid a separate vote on the politically unpopular Senate version by deeming it passed once they approve a package of fixes — has become a focus of Republican attacks this week. And the GOP sought to put Democrats on record defending the maneuver on Thursday with a privileged resolution that would have forced an up-or-down vote on the Senate bill. Democrats turned it back, but not without some heartburn in moderate ranks. And aides said the procedural path forward remained an open question after Republicans appeared to have connected with their withering criticism of the “deem and pass” method.

Steven T. Dennis contributed to this report.