\documentclass[tikz]{standalone}

\usepackage{tikz}

\begin{document}
\usetikzlibrary{automata,positioning}

\iffalse
  Let $M = $ ACACT be a motif.
  $\Sigma = \{ACT\}$
  \fi
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid, auto]
% TODO
  \node[state, initial] (q_0) {$q_{0}$};
  \node[state,right=of q_0] (q_1) {$q_{1}$};
  \node[state,right=of q_1] (q_2) {$q_{2}$};
  \node[state,right=of q_2] (q_3) {$q_{3}$};
   \node[state,right=of q_3] (q_4) {$q_{4}$};
   \node[state,right=of q_4,accepting] (q_5) {$q_{5}$};

  % M itself
  \path[->] (q_0) edge node {A} (q_1)
  (q_1) edge node {C} (q_2)
  (q_2) edge node {A} (q_3)
  (q_3) edge node {C} (q_4)
  (q_4) edge node {T} (q_5);


  % Make it complete so that it recognize all motif in the sequence
  \path[->] (q_0) edge[loop above] node {C, T} (q_1);
  \path[->] (q_1) edge[loop above] node {A} (q_1);
  \path[->] (q_1) edge[bend left=30] node {T} (q_0);
  \path[->] (q_2) edge[bend left=30] node {C, T} (q_0);
  \path[->] (q_3) edge[bend right=30] node[above] {A} (q_1);
  \path[->] (q_3) edge[bend left=40] node {T} (q_0);
  \path[->] (q_4) edge[bend right=50] node[above] {A} (q_3);
  \path[->] (q_4) edge[bend left=40] node {C} (q_0);
  \path[->] (q_5) edge[bend right=40] node[above] {A} (q_1);
  \path[->] (q_5) edge[bend left=50] node {C,T} (q_0);
\end{tikzpicture}

\end{document}