Solving the Secondary Structure Matching Problem in Cryo-EM De Novo Modeling Using a Constrained $K$-Shortest Path Graph Algorithm

Electron cryomicroscopy is becoming a major experimental technique in solving the structures of large molecular assemblies. More and more three-dimensional images have been obtained at the medium resolutions between 5 and
$10 \AA$
. At this resolution range, major
$\alpha$
-helices can be detected as cylindrical sticks and β-sheets can be detected as plain-like regions. A critical question in de novo modeling from cryo-EM images is to determine the match between the detected secondary structures from the image and those on the protein sequence. We formulate this matching problem into a constrained graph problem and present an
$O({\rm \Delta }^2 N^2 2^N)$
algorithm to this NP-Hard problem. The algorithm incorporates the dynamic programming approach into a constrained
$K$
-shortest path algorithm. Our method, DP-TOSS, has been tested using α-proteins with maximum 33 helices and
$\alpha{\hbox{-}}\beta$
proteins up to five helices and 12
$\beta$
-strands. The correct match was ranked within the top 35 for 19 of the 20 α-proteins and all nine
$\alpha {\hbox{-}}\beta$
proteins tested. The results demonstrate that DP-TOSS improves accuracy, time and memory space in deriving the topologies of the secondary structure elements for proteins with a large number of secondary structures and a complex skeleton.