How to investigate a memory leak with Apache and PHP?

We know the memory problem is coming from apache/PHP because whenever we issue a /etc/init.d/httpd reload the memory usage drops

No - that just means it's related to the web traffic. You've gone on to mention that you're running mysql on the box - presumably managing data for the webserver - it could just as easily be the culprit here. As could other services your webstack uses which you've not mentioned.

Each apache thread is assigned a PHP memory_limit of 512MB which explains

No it doesn't. You're reporting an average of 7 and a max of 25 busy servers - yet your memory graph shows a delta of around 25Gb.

Really you should start again with basic HTTP tuning - you seem to be running a constant 256 httpds, yet your peak usage is 25 - this is just plain dumb.

and a max_execution_time of 120 sec which should terminate threads which execution is taking longer

No - only if the thread of execution is within the PHP interpreter - not if PHP is blocked.

that performs financial modeling

(sigh)

It would have been helpful if you'd provided details of how you have configured Apache, threaded or prefork, what version, how PHP is invoked (module, cgi, fastcgi), whether you are using persistent connections, whether you use stored procedures.

I'd suggest you start by moving mysql onto a seperate machine and stop using persistent connections (if you're currently using them). Set the memory limit much lower and override this on a per-script basis. Make sure you've got the circular reference garbage collector installed and configured.

You probably solved your problem by now. As an interim to keep the server from swapping / thrashing I run the following command every hour from cron:

#!/bin/sh 
sync; echo 3 > /proc/sys/vm/drop_caches

I am not saying this is a solution, just a way to keep things running and to minimize downtimw as you investugate the actual cause of the memory leak.

More details can be found here.

http://www.tecmint.com/clear-ram-memory-cache-buffer-and-swap-space-on-linux/

Apparently this is the way PHP works - and if you are doing long loops where you are allocating objects and who knows if you are passing them also via reference, so the only way to deal with it, is to after N requests for each PHP process to stop it. If you run PHP as CGI, every request makes it respawning - so no memory leak, and the performance drop might not be so big. You can also run fast-cgi, where e.g. each 1000 requests the php-fcgi process is killed and it's memory released - again no memory leak. If you run PHP as module mod_php, you might try to setup maxrequests in httpd.conf to see if it helps. I would try to setup e.g. 10 - if it's going to work, the performance drop will not be high, but there should be no memory leaks, even under heavy spike when all 250 httpds are in use (10*250 = 2500 - for each 10MB memory usage is 25GB - so maybe if you dont have 128GB RAM try also to lower the httpd number of processes to e.g. 50).