  A path $P$ in an edge-colored graph $G$ is a \emph{proper path} if no two
adjacent edges of $P$ are colored with the same color. The graph $G$ is
\emph{proper connected} if, between every pair of vertices, there exists a
proper path in $G$. The \emph{proper connection number} $pc(G)$ of a connected
graph $G$ is defined as the minimum number of colors to make $G$ proper
connected. In this paper, we study the degree sum condition for a general graph
or a bipartite graph to have proper connection number 2. First, we show that if
$G$ is a connected noncomplete graph of order $n\geq 5$ such that
$d(x)+d(y)\geq \frac{n}{2}$ for every pair of nonadjacent vertices $x,y\in
V(G)$, then $pc(G)=2$ except for three small graphs on 6, 7 and 8 vertices. In
addition, we obtain that if $G$ is a connected bipartite graph of order $n\geq
4$ such that $d(x)+d(y)\geq \frac{n+6}{4}$ for every pair of nonadjacent
vertices $x,y\in V(G)$, then $pc(G)=2$. Examples are given to show that the
above conditions are best possible.
