Will Joe Biden run? It's one of the biggest questions hanging over the Democratic presidential primary race right now, and a new Washington Post-ABC News poll illustrates one reason why.

Biden wins support from about 1 in 5 Democratic-leaning voters (21 percent) in the poll, up from 12 percent in July as news about his potential candidacy has bubbled. Biden's rise paralleled a 10-point jump in support for Bernie Sanders (to 24 percent) and a 21-point drop for Clinton. And despite the similar rises of Sanders and Biden, the numbers suggest Biden's candidacy would prove much more harmful to Clinton than the Vermont senator.

This is not hugely surprising (we've examined this here and here), but the poll allows for a fine-grained look at the Biden Factor; it asked Biden supporters who their second-choice candidate is, making it possible to construct a race with and without Biden included. The graphic below shows support among Democratic voters with and without Biden in the race.

Clinton is hurt most by Biden's inclusion, dropping 14 percentage points (from 56 percent without Biden included to 42 percent with Biden). Sanders also gains with Biden out of the race, but by a much-smaller four percentage points. Do the math, and Biden's entry makes the difference between a 28-point lead for Clinton and an 18-point lead. And given the increasingly close race, those 10 points could wind up mattering.

One reason for this dynamic might be that Biden and Clinton supporters are similar ideologically and demographically. Sample sizes are too small in an individual poll to analyze the makeup of Biden's and Sanders's supporters, so we've combined our July and September Post-ABC polls with results among voting-age Democrats to see how each candidate's supporters line up.

Political ideology illustrates the clearest similarity between Biden and Clinton backers; 33 percent of Biden's supporters and 41 percent of Clinton supporters are liberal, but that rises to 62 percent among those who support Sanders. Biden's backers are also closer to Clinton in the share who are non-white, the share who graduated from college and the share who are women (caveat: Sanders's support among women did grow sharply from July to August). On the flip side, Biden's voters were roughly in the middle of Clinton and Sanders supporters when it comes to young voters (those aged 18-to-39) and political independents.

Altogether, the polls show Biden's support draws from some similar parts of the Democratic party as Clinton, making her the candidate with the most to lose if he runs. This makes sense, given that Biden is not expected to run as a liberal's liberal like Sanders. It also indicates Sanders's heavily liberal support might be more insulated to a Biden run, potentially helping him prevail with a plurality -- rather than a majority -- of the vote in primaries and caucuses where Clinton and Biden compete for moderates, conservatives and voters of color in the party.

Now, there are plenty of other possible ways a Biden run could help or hurt Clinton in the long run. Sean Trende at Real Clear Politics delved into some scenarios earlier this month, arguing that a Biden entry might upend a news narrative focused heavily on Clinton's surprising struggle and that her campaigning quality might improve from additional competition and that winning against both Biden and Sanders will be seen as a greater accomplishment.

So a Biden entry might not be all terrible for Clinton, but she'd rather just have his voters' support in the primary.

Peyton M. Craighill contributed to this post.

Methodology: The Washington Post-ABC News poll was conducted by telephone Sept. 7-10 among a random national sample of 1,003 adults, including interviews in English and Spanish on conventional and cellular phones. Results among the sample of 356 Democrats and Democratic-leaning independent registered voters have a margin of sampling error of plus or minus six percentage points. The combined September/July results are among voting-age Democrats, with error margin of five points among the sample of 484 Clinton supporters and nine points for the sample of 155 Sanders supporters and 153 Biden supporters. See here for more detail on the survey's methodology.