Try
While I agree with the other answers, I feel they're missing one important aspect - trying.
While there may be little chance of success with a bit of an attempt, there's even less risk involved - as long as you're not too forceful or persistent, it's unlikely to significantly change their opinion of you (unless it's positively).
The cause
But first, note that an important aspect here is what the cause of your below-expectation performance was. Or, more importantly, what they think the cause was.
If the cause was, for example, lacking knowledge required to do this job, they might be able to test this in an interview setting (assuming they believe they can do so better than they did the first time around), and you may have success if you try. And the same applies if you're applying to a different type of job.
If, on the other hand, it was caused by (or they think it was caused by), for example, laziness, you're less likely to have much luck.
Back to trying
Now onto the actual attempt.
This may differ significantly based on the method they used to contact you.
If by e-mail (and you haven't replied yet), this is probably the easiest.
You can probably reply with something like:
I appreciate your honesty.
I believe I have matured a lot and have learnt a lot about {whatever you do} in the last 4 years and have made significant contributions to {whatever domain you work in} (which I firmly believe my references will agree with) and I would greatly appreciate being given another consideration.
However, I would respect your decision to not reconsider my application, if you choose not to do so.
(obviously along with appropriate headers and footers and customized to take into account whatever factors contributed to you having been laid off, but don't make it too long, and don't just copy the above - you should be able to justify any statements made)
Disclaimer: I'm not exactly the most skilled with language.
If contacted by phone or you've already replied, it could be a bit more difficult.
Closing note
From the company's perspective - people don't typically change that much in 4 years (although they can learn quite a bit). You can always try again in a few more years, if you're still interested.