I have a relatively old site, with a fairly good pagerank, that I'm in the process of upgrading. The second-tier pages are also well pageranked. After reading through the content on these pages, I realize that their content needs to be completely overhauled and re-written. Is this content change likely to affect the pagerank of those URLs (negatively or positively)?
Overhauling a website is fine but if pages are ranking you should not break them and should try to keep certain things in place that made the ranking possible in the first place. Removing key elements will most likely drop your page temporarly or perm. Google likes aged pages, and changes are good but again don't break it. You should not touch key elements of the page and add around them, these key elements are Page Title, Meta Description, h1 h2 h3 h4 h5 h6 tags. Obey this and you should see a improvement or no change in your rankings. Other changing to much can be damanging and I'll mention why later.
Many people are still in the 1990's / early 2000's mindset that page rank is important, page rank is a very minor factor these days and should not even be considered when overhauling your site. A PR0 site can beat a lot of high PRs if the authorithy of the site is good.
PR is controlled by Google and only they know the factors which will result in increase of decrease, but most of it PR is backlinks, age of page, age of links and quality of page. of which you should be fine in both cases.
Search engines now work on Authority and how relevant your in and out going links are. For example you could have a PR7 site linking to you and make your site PR5 however because your site is about Dogs, and the PR7 site is about Airplanes you will gain little to non juice from that site - Clearly showing you that PR inst a reliable measurement how good a site is.
However PR can be a good indicator if their link network is CLEAN. For example if they have PR4 and the site is about Cats, and all their links to them is from other cats sites then yes, they will rank very well! and this is a good measurement - but you need to check the links before deciding if its a good measurement or not, which so many do not.
Why Should You Not Change Key Elements
There are many reasons why you should not change the key elements but most importantly its because Google doesn't want people changing things just to rank better, if you go changing H1, Metas and so forth on all the pages its obvious that you want to rank the pages higher, Google has things in place to moniter page changes, they have many patents that detects key elements page changes. They don't want people having the ability to know what ranks better than this and that, so in most cases changing to much at once normally results in negative results. If you want to change H1, Titles, Metas and so forth its better to do it over a set period of time.
PageRank? No. Page ranking? Yes.
PageRank is a numerical representation of a page's link popularity. It is not directly related to page content but is only directly affected by incoming links.
A page's rank is where it ranks in the search results. The content on a page is very much a factor in that and a rewrite will almost certain change the rankings. You won't know if it's for the better until after you make the changes and see what happens.
your question is quite clearly answered here. Both the seems to be same question, only change you are asking about effect on pagerank and in given question(below) it is asking about SEO effect.