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GNM with Protein

GNM has already been thoroughly and successfully applied to proteins. There is even a database known as iGNM that has calculated protein fluctuations for over 20,000 structures from the protein data bank or PDB. In most cases, carbon atoms are chosen as the nodes to represent the network. A common cutoff value is 7 Angstroms. To show you that GNM works, in Figure 1, we calculated the fluctuation for HIV-1 protease and compared it to the temperature factor, which are experimental values. There are 200 nodes in this structure.

Ivet Bahar et al.

Figure 1: (a)Fluctuation data for HIV-1 protease predicted by GNM and compared with temperature factor (TF) (b)Original structure (c)Colored fluctuation structure

Qualitatively, it is very easy to see that the GNM results (shown in red) match very well to the TF results (shown in black). Each dot represents a node and lines are simply to help guide your eyes. GNM has been successful in calculating protein fluctuations because of well matching results such as the one in Figure 1.

GNM with DNA

There haven’t been many applications of GNM on DNA structures. But since GNM filters out the chemistry of the atoms and simply looks at the bonds between the nodes, there is no reason why it shouldn’t work. In Figure 2, we tried GNM with DNA of PDB ID 307D. We used the phosphate atoms as the nodes and a cutoff distance of 13 Angstroms. There are 60 nodes. Results are pretty satisfactory but are slightly less accurate than the comparison with protein. This is because proteins have longer chains than DNA and more nodes, thus it rules out randomness. DNA is more flexible and thus harder to calculate the flexibility. We chose 307D strictly because it had a long chain.


Figure 2: (a)Fluctuation data for 307D (PDB ID) predicted by GNM and compared with temperature factor (TF) (b)Original structure (c)Colored fluctuation structure

Again, DNA fluctuation results from GNM (in red) are compared to TF results (in black). Points represent nodes and lines are to guide your eyes.

GNM with Synthetic DNA

Now that we have shown that GNM is applicable in both proteins and DNA, we want to use it to calculate fluctuations of synthetic DNA. Below you can see 2 structures that we've made using NanoEngineer-1, a cube and a prism. High fluctuations are shown in warm colors (red, orange, yellow/yellow-green) and low fluctuations are shown in cool colors (green, blue, dark blue). Both structures show similar satisfactory results. It makes sense to think that the vertices of the structure are the least flexible because they are the most confined. The legs of the prism are more flexible because they are not as rigidly confined.


Figure 3: Fluctuation data of a cube (top) and prism (bottom) structure created in NanoEngineer-1 and calculated using GNM.

To verify that the GNM method gives accurate results to synthetic DNA structures, we compared our data with another method, Finite Element Method (FEM). Figure 4, left, you can see a half-gear structure that we created in NanoEngineer-1 and calculated using GNM. On the right is a similar structure from Dietz et al.[1] with calculation done by CanDo [2] using FEM. There is clearly similarities between these two structures. Both show highly fluctuating ends and teeth (in red/yellow/yellow green) and less flexible central body (in blue/dark blue). Note that there are a few differences between the results from GNM and FEM. We hand built the structure in NanoEngineer-1 so there is always some variations from the FEM version. Additionally, NanoEngineer-1 could not export temperature factors in the PDB files. Without these temperature factors, we could not use common biochemistry graphics software to generate better color-based flexibility for our figures.

Compare gnm.png

Figure 4: Verification of the results by comparison to Finite Element Method (FEM).


There are multiple advantages of our GNM method. They include:

  • Fast calculation time: results take from seconds to minutes depending on size.
  • Large deformations: there is a solution to every problem. Because GNM only considers the bonds and distances between nodes, it can calculate fluctuations for even large deformations in the structure.
  • Physical relevance: the diagonal of the Kirchhoff matrix corresponds to fluctuation of the structure. Each value corresponds to a node which in turn represents an atom within the structure.
  • Feedback ability: explained below.


Structural feedback is one of the most advantageous parts of our GNM method. It allows one to modify the structure fluctuations on a custom level. Figure 5 below displays this aspect of our method. Figure 5(d) shows the original structure. One may think that each leg of the structure should have even flexibility because it is symmetric. However, this is not the case as the 2 front legs seem to have a higher flexibility then the back legs and even their fluctuations are not the same. The reason for this is because these structures were built by hand. There will always some errors in the structures that cause these variations. Figure 5(b) on the other hand, shows the structure after feedback has been applied to it. Staples were added to the front legs thus stabilizing the structure (noting that warm colors are high fluctuation and cool colors are low fluctuation). Staples are simply a bond between one double stranded DNA strand to the next. Adding staples to the DNA structure introduces confinement to the strands, which in turns, decreases the flexibility. Figure 5(c) shows a comparison of the fluctuation data from before and after staple confinement. The red data shows structure flexibilities after staples were added while the black shows flexibilities from the original structure. Each point represents a node and the lines are simply to guide your eyes. It can be seen that the fluctuations are lessened after staples are added, emphasizing the point that staples within the structure help to stabilize and can be used to customize structure properties. Figure 5(a) is simply to show what the structure appeared like in NanoEngineer-1 for clarity.


Figure 5: Feedback shown on the cube structure. a)Ribbon representation of the structure in NanoEngineer-1. b)MATLAB color-coded 3D graph of the structure after staples were added. c)Fluctuation comparison of before (in black) and after (in red) staples were added. d)MATLAB color-coded 3D graph of the original structure before any modifications.

Figure 6 further shows how feedback can be applied on a customized basis. In a hypothetical situation in which you would want one leg to be strictly confined but the other leg free, our GNM method can be used to create just this case. By adding staples only to one leg, one can purposefully create a structure that has variations in the flexibility. This also shows that adding staples would only add confinement on a regional basis. Nodes that are outside of the cutoff distance are still not affect by the staple.

Feedback 1leg.png

Figure 6: Feedback applied on a custom basis. One leg is left without staples for extra flexibility.

Figure 7 shows a similar result with the prism structure. 7(a) shows the structure created in NanoEngineer-1. 7(b) shows the structure after staples were applied to all the legs to add confinement and decrease flexibility. This is show by a color change to green. 7(d) is the original structure before staples were added, shown for comparison to 7(b). 7(c) is a comparison of the fluctuation graph from before(black) and after staples(red) were added.

Feedback prism.png

Figure 7: Feedback shown on the prism structure. a)Ribbon representation of the structure in NanoEngineer-1. b)MATLAB color-coded 3D graph of the structure after staples were added. c)Fluctuation comparison of before (in black) and after (in red) staples were added. d)MATLAB color-coded 3D graph of the original structure before any modifications.


Structure files: Media: NE1 structures.rar