Integers a, b, c, d, e

As $a$ , $b$ , $c$ , $d$ , and $e$ are distinct integers, $(4-a)$ , $(4-b)$ , $(4-c)$ , $(4-d)$ , and $(4-e)$ must be distinct integers.

There is only one such case where $12$ has $5$ distinct factors that are integers;

$12=(3)(2)(1)(-1)(-2)$

So,

$4-a=3 \implies a=1$

$4-b=2 \implies b=2$

$4-c=1 \implies c=3$

$4-d=-1 \implies d=5$

$4-e=-2 \implies e=6$

Therefore, $a+b+c+d+e = 1+2+3+5+6 = \boxed{17}$