4. Integers $a, b, c$ satisfy $a+b+c=2$, and
$$
S=(2 a+b c)(2 b+c a)(2 c+a b)>200 \text {. }
$$

Then the minimum value of $S$ is $\qquad$.
4. 256.

Since $a+b+c=2$, we have
$$
\begin{aligned}
S= & (b c-2 b-2 c+4)(c a-2 c-2 a+4) . \\
& (a b-2 a-2 b+4) \\
= & (b-2)(c-2)(c-2)(a-2) . \\
& (a-2)(b-2) \\
= & (a-2)^{2}(b-2)^{2}(c-2)^{2}
\end{aligned}
$$

is a perfect square.
Also, since $a+b+c=2$ and $a, b, c$ are integers, at least one of $a, b, c$ is even.
Thus, $S$ is even.
Given $S>200$, we have
$$
S \geqslant 16^{2}=256.
$$

When $a=4, b=0, c=-2$, $S=256$.
Therefore, the minimum value of $S$ is 256.