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WIREs Comput Mol Sci
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# Rigidity theory for biomolecules: concepts, software, and applications

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The mechanical heterogeneity of biomolecular structures is intimately linked to their diverse biological functions. Applying rigidity theory to biomolecules identifies this heterogeneous composition of flexible and rigid regions, which can aid in the understanding of biomolecular stability and long‐ranged information transfer through biomolecules, and yield valuable information for rational drug design and protein engineering. We review fundamental concepts in rigidity theory, ways to represent biomolecules as constraint networks, and methodological and algorithmic developments for analyzing such networks and linking the results to biomolecular function. Software packages for performing rigidity analyses on biomolecules in an efficient, automated way are described, as are rigidity analyses on biomolecules including the ribosome, viruses, or transmembrane proteins. The analyses address questions of allosteric mechanisms, mutation effects on (thermo‐)stability, protein (un‐)folding, and coarse‐graining of biomolecules. We advocate that the application of rigidity theory to biomolecules has matured in such a way that it could be broadly applied as a computational biophysical method to scrutinize biomolecular function from a structure‐based point of view and to complement approaches focused on biomolecular dynamics. We discuss possibilities to improve constraint network representations and to perform large‐scale and prospective studies. WIREs Comput Mol Sci 2017, 7:e1311. doi: 10.1002/wcms.1311

• Structure and Mechanism > Computational Biochemistry and Biophysics
• Computer and Information Science > Computer Algorithms and Programming
• Software > Molecular Modeling
Schematic representation of a structural engineering construction (bridge) consisting of struts (distance constraints) connected by joints. (a) In 2D, the triangle is the smallest rigid unit. Hence, if all constraints are in place, the bridge is isostatic or minimally rigid. (b) Removing one constraint divides the bridge into two rigid clusters with a flexible region in between.
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Allosteric pathways in the ribosomal exit tunnel. (a) Rigid cluster decomposition of the allosteric pathway to the peptidyl transferase center (PTC) (red) as predicted by Fulle et al. Different shades of blue correspond to different rigid clusters. Residues in orange were identified to be important for ribosome stalling in experiments. Figure adapted from Ref . (b) Allosteric pathways for PTC silencing (R1, R2, R3) when the tryptophanase C (TnaC) peptide (green) is in the exit tunnel; the grey loops marked L4 and L22 indicate ribosomal proteins. Residues that agree with the prediction of the rigidity analyses from (a) are colored accordingly and circled in red. Ribosomal components not identified in the rigidity analysis are colored in grey. Orange residues as in (a). Figure adapted from Ref .
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Long‐range coupling effects in RNA and protein. (a) Schematic representation of long‐range allosteric coupling in the protein tyrosine phosphatase 1B (PTP1B). Upon perturbing the network at the allosteric site by adding constraints mimicking the binding of an allosteric modulator (red), altered stability characteristics are observed for the functionally important WPD loop (orange) and for residues in the orthosteric site (green). (b) Schematic representation of the long‐range cooperative stabilization of the P1 region in the aptamer domain of the guanine‐sensing riboswitch. Interactions within the tertiary loop‐loop region (red) and of the ligand with the binding site (red) together are required to stabilize the terminal P1 region (green) (C.A. Hanke, H. Gohlke, unpublished results).
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Application of rigidity theory to investigate protein thermostability. (a) Correlation between $r c ˜$ ij,neighbor , a local measure for predicting thermodynamic stability, and experimental thermostabilities (T m) for the six wild type crystal structures (empty squares) and thirteen variants of the Bacillus subtilis lipase A. For the six wild type crystal structures, the resulting mean $r c ˜$ ij,neighbor is shown as a horizontal bar. Experimental values were taken from Refs . Error bars depict the standard error in the mean. (b) Stability map of the variant 6B, T m of which is 6.6 K higher than that of the wild type. A red/blue color shows that a rigid contact in the variant is more/less stable than in the wild type (see color scale). The upper triangle shows differences in stability values for all residue pairs, and the lower triangle shows differences in stability values only for residue pairs that are within 5 Å of each other. Secondary structure elements are indicated on both abscissa and ordinate and are labeled: α‐helix (red rectangle), β‐strand (green rectangle), loop (black line). Arrows represent the mutation positions with respect to the wild type sequence. (c, d) Structures of the variants 6B (c) and 1–14F5 (d); T m of 1–14F5 is 2.1 K higher than that of the wild type. Common mutations in 6B and 1–14F5 are shown in magenta, unique mutations in 6B are shown in green. The differences in the stability of rigid contacts for residue neighbors is displayed by sticks connecting Cα atoms of residue pairs colored according to the scale shown in panel (b); only those contacts that are stabilized by ≥ 4 K or destabilized by ≥ 3 K are shown for clarity. Figure adapted from Ref .
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Overview of the constraint network types, algorithms, and software packages discussed in this review.
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Results of a constraint dilution simulation of hen egg white lysozyme with CNA. (a) In the constraint dilution simulation, a stepwise decrease in the cutoff energy (E cut) removes hydrogen bonds from the constraint network in the order of increasing strength. The colored surfaces represent the rigid clusters, and the black lines represent the flexible regions of the protein. (b) Degree of disorder along a constraint dilution simulation as revealed form the cluster configuration entropy H. The disorder is low when a single rigid cluster dominates and increases when the cluster falls apart into smaller subclusters of different sizes. (c) The rigidity index ri characterizes the per‐residue stability as it monitors when a residue i segregates from any rigid cluster during a constraint dilution simulation. A lower ri value indicates that the residue resides in a region of higher stability. (d) Stability maps (upper triangle) and neighbor stability maps (lower triangle) represent when a ‘rigid contact’ between two residues of the network (both residues belong to the same rigid cluster) vanishes during the constraint dilution simulation. Gray areas in the neighbor stability map indicate that no native contact exists for that residue pair. Figure adapted from Ref . Note that arrows at axes labeled with Ecut point in the direction of more negative values.
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The 3D pebble game algorithm; see Box 2 for details. Figure adapted from Ref .
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Double banana network. Constraint counting implies that the 3D double banana network is rigid because it satisfies the 3N − 6 counting condition considering that the nodes have three DOF. However, internal motion within this network is possible along the implied‐hinge joint between the two ‘banana’ subgraphs (dashed line). Figure adapted from Ref .
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Modeling of covalent and noncovalent interactions. For both (a) interactions within a protein and (b) RNA, the rigid clusters (green) and overconstrained regions (blue) are shown. For rigidity analysis, covalent interactions (black lines), hydrogen bonds (yellow squared dots) and salt bridges (yellow hatched lines), and hydrophobic interactions (cyan squared dots) are modeled as constraints. For RNA also base‐stacking interactions (cyan hatched lines) are modeled as hydrophobic interactions.
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Constraint network representations. (a) Ball‐and‐stick representation of propene, the carbon atoms are shown in blue and the hydrogen atoms in light gray. (b, c, d) Propene is represented in terms of 3D constraint networks. (b) In the bond‐bending network (also called bar‐and‐joint network or molecular framework) covalent bonds are modeled as distance constraints between nearest‐neighbor atoms (thick lines) and angle constraints between next‐nearest‐neighbor atoms (dashed lines). For the double bond (c,d), there is an additional constraint (red dotted line) between third‐nearest‐neighbor nodes (b,f), removing the bond‐rotational DOF between the two sp2 carbons. The network represented here has a total of nine nodes, connected by eight distance constraints, eleven next‐nearest‐neighbor constraints, and one third‐nearest‐neighbor constraint. In this network, a node (atom) has three DOF, leading to a 3N − 6 count (Eq. (1) in Box ). With N = 9 nodes and a total of 20 nonredundant constraints, this network has one DOF, the rotation around the single bond. (c) In the body‐and‐bar representation, atoms are modeled as bodies with six DOF, a covalent single bond as five constraints between two bodies, and a double bond as six constraints. (d) In the body‐bar‐hinge model, all covalent bonds are replaced by hinge regions, located at the connection of two colored shapes, connected in such a way that one DOF is left. For the double bond, an additional bar (red dotted line) is added to the hinge region to lock the remaining DOF. (e) The modeling of bond types is compared between the bond‐bending network (left column) and the body‐and‐bar network (right column): The covalent bond with five constraints (top), the double bond with six constraints (middle), and the hydrophobic interaction modeled with ghost atoms in the bond‐bending network (bottom left) and with two bars in the body‐and‐bar network (bottom right). Figure 2(e) adapted from Ref .
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