Normal mode analysis

^{by Robert Greil and Cedric Landerer}

Introduction

NMA (normal mode analysis) is a time-independent approach to simulate low-frequency motions and vibrations of protein. These simulation are all based on the harmonic approximation and therefore ignore the influence of the solvent. The proteins are seen as models made out of springs and point masses, which are connected and represent the interatomic forces. Simulation done this way are very easy to do, but are no more than a slight insight into the protein flexibility.

Network Models

The Normal mode analysis is done by two different types of network models: The Elastic Network Model (ENM) and the Gaussian Network Model (GNM).

The ENM belongs to the standard NMA but all springs are considered of equal strength and only the atom pairs below an threshold are used.

The GNM uses masses of points to represent the Cα atoms of the proteins and model the interaction between those point masses with harmonic approximation.

WEBnm@

WEBnm@<ref name=webnma>Siv Midtun Hollup, Gisle Salensminde and Nathalie Reuter. "WEBnm@: a web application for normal mode analyses of proteins". BMC Bioinformatics 2005</ref> is a webserver based application that allows computation and low-frequency analysis of normal nodes of proteins. This computation is fully automated and only different types of results are presented to the user.

Usage

• Webserver:
  • http://apps.cbu.uib.no/webnma/home
• Input:
  • PDB: 1a6z
  • Chains: all
• Output:
  • Animation of harmonic movements
  • Plot of squared atomic displacements
  • Energy of the deformation

Result:

Figure 1.1: Mode 7 by WEBnm@: plot

Figure 1.2: Mode 7 by WEBnm@: vibrations

Figure 2.1: Mode 8 by WEBnm@: plot

Figure 2.2: Mode 8 by WEBnm@: vibrations

Figure 3.1: Mode 9 by WEBnm@: plot

Figure 3.2: Mode 9 by WEBnm@: vibrations

Figure 4.1: Mode 10 by WEBnm@: plot

Figure 4.2: Mode 10 by WEBnm@: vibrations

Figure 5.1: Mode 11 by WEBnm@: plot

Figure 5.2: Mode 11 by WEBnm@: vibrations

Figure 6.1: Mode 12 by WEBnm@: plot

Figure 6.2: Mode 12 by WEBnm@: vibrations

Discussion:

All animated gifs have to be created the hard way, frame after frame, because WEBnm@ does not allow the concurrent saving of more than one frame.

The Normalized Squared Atomic Displacements (nsad) plots show the vibrations according to the amino acid position. The peaks are almost always at the beta-sheets of chains A and C.

Except for mode 11 there is no special movement inside the alpha helix of chains A and C. The movement is almost everytime between the chains or inside/around the beta strands of chain B and D. This behavior is also visible by analyzing the plots; the regions of low movement are always around the chains A and C with their corresponding alpha helices and the high movement regions lies within the beta strands of chain B and D.

The movement/vibrations can be described mostly as repulsive or flattening, stretching and twisting.

ElNemo

ElNémo<ref name=elnemo>Karsten Suhre and Yves-Henri Sanejouand. "ElNémo: a normal mode web server for protein movement analysis and the generation of templates for molecular replacement". Nucleic Acids Res. 2004 Jul 1;32(Web Server issue)</ref> is a webserver based to work with the Elastic Network Model. It calculates and analyses low-frequency normal modes of proteins with no restriction because it uses the building block approximation.

Usage

• Webserver:
  • http://www.igs.cnrs-mrs.fr/elnemo/start.html
• Input:
  • PDB: 1a6z
  • Amount of modes: 20 (trivial modes 1-6 are not available)
  • Threshold for elastic interactions
• Output:
  • Animation of the movements
  • Plot of mean square displacement of Cα atoms
  • Fluctuations of the distance between Cα atoms

Result:

Figure 7.1: Mode 7 by ElNemo: lateral view

Figure 7.2: Mode 7 by ElNemo: top view

Figure 7.3: Mode 7 by ElNemo: front view

Figure 8.1: Mode 8 by ElNemo: lateral view

Figure 8.2: Mode 8 by ElNemo: top view

Figure 8.3: Mode 8 by ElNemo: front view

Figure 9.1: Mode 9 by ElNemo: lateral view

Figure 9.2: Mode 9 by ElNemo: top view

Figure 9.3: Mode 9 by ElNemo: front view

Figure 10.1: Mode 10 by ElNemo: lateral view

Figure 10.2: Mode 10 by ElNemo: top view

Figure 10.3: Mode 10 by ElNemo: front view

Figure 11.1: Mode 11 by ElNemo: lateral view

Figure 11.2: Mode 11 by ElNemo: top view

Figure 11.3: Mode 11 by ElNemo: front view

Discussion:

For all generated models the vibrations are shown in three different perspectives.

The Movement/Vibrations are very similar to these obtained by WEBnm@. There is almost no movement inside the alpha helices of chain A and C and much movement inside and outside the the beta strands of chain B and D. Vibrations between chains can also be observed but these are mostly between A+B and C+D because they form a subunit.

Anisotropic Network Model web server

The Anisotropic Network Model (ANM)<ref name=anm>Atilgan AR, Durell SR, Jernigan RL, Demirel MC, Keskin O, Bahar I. "Anisotropy of fluctuation dynamics of proteins with an elastic network model.". Biophys J. 2001 Jan;80(1)]</ref><ref name=anm_srv>Eyal E, Yang LW, Bahar I. "Anisotropic network model: systematic evaluation and a new web interface.". Bioinformatics. 2006 Nov 1;22(21)</ref> web server uses the fast approach anisotropic network model (elastic network) to calculate the global modes and all spring forces are based on the constant γ.

Usage

• Webserver:
  • http://ignmtest.ccbb.pitt.edu/cgi-bin/anm/anm1.cgi
• Input:
  • PDB: 1a6z
  • Threshold for interactions
  • Distance weight for interactions
• Output:
  • Animation of movements
  • Deformation energies
  • Profiles of fluctuation
  • Residue distance fluctuations

Result:

Figure 12: Mode 1 by ANM

Figure 13: Mode 2 by ANM

Figure 14: Mode 3 by ANM

Figure 15: Mode 4 by ANM

Figure 16: Mode 5 by ANM

Figure 17: Mode 6 by ANM

Discussion:

All models created by ANM have something in common: The flexible parts are the outer beta-sheets and the inflexible or rigid parts are the alpha-helices inside the protein. The movements are therefore mostly the same, the outer parts move in some directions and the inner parts stand still or are moved coercively.

Figure 12 shows movement of beta-sheets from back to front and thus slightly rotating the inside of the protein between the A and C chain.

In figure 13 the other beta-sheets are going up and so the inner is moved a bit down as a follow-up reaction.

Figure 14 is a nice one, because there is rotation inside the chains A and C. The connection between the alpha-helices and beta-sheets is moved to the top while the chains B and D are moved to the bottom, thus letting the protein wobble.

The mutual vibration of the outer beta sheets (up and down) of figure 15 create a shaking inside the protein. But the alpha-helices are not moved, they are still fixed but only tipped to left an right.

The outer parts are again in mutual vibration. This time they are pulled to the center of the protein at the front and at the back at same time. That induces a turning of the alpha-helices in the middle of the protein as seen in figure 16.

The last figure 17 shows the outer parts being lifted up and the whole protein looks compressed.

oGNM – Gaussian network model

The oGNM<ref name=ogn>iGNM: A Database of Protein Functional Motions Based on Gaussian Network Model." Lee Wei Yang, Xiong Liu, Christopher Jon Jursa, Mark Holliman, A.J. Rader, Hassan Karimi, Ivet Bahar. Bioinformatics Jul 2005</ref>is an online web-server that calculates the dynamics of PDB structures using an elastic network model which is called gaussian network model.

Usage

• Webserver:
  • http://ignm.ccbb.pitt.edu/GNM_Online_Calculation-t.htm
• Input:
  • PDB: 1a6z
  • Number of nodes
  • Threshold for amino acid and nucleotide pairs
  • Visualization type
• Output:
  • Blue-red painted pictures of protein mobility
  • Plot of fluctuations per residue
  • Plot of cross-correlation of different modes

Result:

Figure 18.1: Mode 1 by oGNM: plot

Figure 18.2: Mode 1 by oGNM: vibrations

Figure 19.1: Mode 2 by oGNM: plot

Figure 19.2: Mode 2 by oGNM: vibrations

Figure 20.1: Mode 3 by oGNM: plot

Figure 20.2: Mode 3 by oGNM: vibrations

Figure 21.1: Mode 4 by oGNM: plot

Figure 21.2: Mode 4 by oGNM: vibrations

Figure 22.1: Mode 5 by oGNM: plot

Figure 22.2: Mode 5 by oGNM: vibrations

Figure 23.1: Mode 6 by oGNM: plot

Figure 23.2: Mode 6 by oGNM: vibrations

Discussion:

For all figures the plot of chain A and C and B and D is always identical, because they are the same chains only placed in opposite direction. A high value of the y-axis codes a high flexibility for that amino acid position, these parts are marked as red. The inflexible parts are colored in blue.

Figure 18.1 shows clearly the inflexible alpha-helices of chains A and C which are getting more and more flexible as the position advances the beta-sheets. These are the most flexible part of the protein alongside the beta-sheets of chains B and D, which are according to the plot not that flexible. This observation is confirmed by the figure 18.2.

Figure 19.1 is somewhat special, because it illustrates that alpha-helices of chains A and C are not only flexible parts but slightly more flexible as the beta-sheets of chains B and D. This is spectacular because all other models tend to define the alpha-helices as rigid parts. Figure 19.2 shows this aspect as the alpha-helices are slightly red but the beta-sheets of chains B and D are full blue.

Figure 20.1 shows a very mixed up model. Only the outer parts of the beta-sheets are defined as flexible all other parts are more or less static parts.

Figure 21.1 is similar to figure 18.1 excluding the mobility of the beta-sheets of chains A and C. These are almost fixed and therefore achieve more blue coloring.

Figure 22.1 is again a very quirky model because the mobility of the outer beta-sheets is reduced to almost nothing but the flexibility of the beta-sheets of chains B and D is greatly raised.

Figure 23.1 is similar to figure 19.1 but more restrictive in the mobility of the beta-sheets. There is no more flexibility at chain B and D but therefore a very versatile core inside the protein consisting of the alpha-helices of chains A and C.

NOMAD-Ref

NOMAD-Ref<ref name=nomadref>Tirion, Monique M. Large Amplitude Elastic Motions in Proteins from a Single-Parameter, Atomic Analysis. Phys Rev Lett, 77, 1905--1908 (1996).</ref> is an online web-server for normal mode calculations using the elastic network model and the Tirion calculation to include all atoms into the calculation process.

Usage

• Webserver:
  • http://lorentz.dynstr.pasteur.fr/nma/submission.php
• Input:
  • PDB: 1a6z
  • Amount of modes
  • Distance weight in Å
  • Threshold for mode calculation in Å
  • aRMSD (average) in Å
• Output:
  • Animation of movements
  • Plot of CRMS per residue

Result:

Figure 24: Mode 7 by NOMAD-Ref

Figure 25: Mode 8 by NOMAD-Ref

Figure 26: Mode 9 by NOMAD-Ref

Figure 27: Mode 10 by NOMAD-Ref

Figure 28: Mode 11 by NOMAD-Ref

Figure 29: Mode 12 by NOMAD-Ref

Discussion:

Figure 24 shows movement between both of the subunits of 1A6Z. There are no other vibrations inside any of the chains, only rotation between both complexes.

Figure 25 visualizes the the flexible beta-sheets of the chains A and C. These are shifted saw-like with the beta-sheets of chains B and D.

The whole protein is stretched at Figure 26. It is clearly visible, that the beta-sheets are much more flexible than the alpha helices which seems to work as springs, trying to keep the protein in a closely packed state.

In Figure 27 there is a rotation between the complexes of chains A+B and C+D and also again some stretching inside the beta-sheets of chains B and D. The movement is somewhat similar to Figure 26.

Figure 28 is also a rotation between the complexes but also inside the complexes. They are rotated at the connection of the alpha-helices to the beta-sheets of chains A and C.

Figure 29 is almost identical to Figure 28.

As a final result, we assume that M2B avoids the protein from collapsing. Without the M2B beta-sheet, the protein is able to converge into a spherical form. Therefore, we propose that a decreased binding affinity will damage the function of the HFE protein by structural deformation.

All-atom NMA using Gromacs on the NOMAD-Ref server

The all-atom NMA using Gromacs is a normal mode calculation which uses force fiels and not elastic networks. It utilizes all atoms and needs therefore much more time.

Usage

• Webserver:
  • http://lorentz.dynstr.pasteur.fr/gromacs/nma_submission.php
• Input:
  • PDB: 1a6z
  • Temperature in K: 600 and 2000
• Output:
  • Animation of movements

• Additional information:
  • We used the given protein 1BPT because our HFE protein (1A6Z) has around 6080 ATOM lines and is therefore too big (limit is 2000 ATOM lines).

Result:

• at 600K

Figure 30: Mode 7 by Gromacs at 600K

Figure 31: Mode 8 by Gromacs at 600K

Figure 32: Mode 9 by Gromacs at 600K

Figure 33: Mode 10 by Gromacs at 600K

Figure 34: Mode 11 by Gromacs at 600K

Figure 35: Mode 12 by Gromacs at 600K

• at 2000K

Figure 36: Mode 7 by Gromacs at 2000K

Figure 37: Mode 8 by Gromacs at 2000K

Figure 38: Mode 9 by Gromacs at 2000K

Figure 39: Mode 10 by Gromacs at 2000K

Figure 40: Mode 11 by Gromacs at 2000K

Figure 41: Mode 12 by Gromacs at 2000K

• with Elastic Network

Figure 42: Mode 7 by NomadRef Elastic Network

Figure 43: Mode 8 by NomadRef Elastic Network

Figure 44: Mode 9 by NomadRef Elastic Network

Figure 45: Mode 10 by NomadRef Elastic Network

Figure 46: Mode 11 by NomadRef Elastic Network

Figure 47: Mode 12 by NomadRef Elastic Network

Discussion:

As one can see, there is no big difference between the movements at 600K and 2000K. The only difference is the range of the vibrations; at 2000K it is slightly more than at 600K which leads to the conclusion that the movements do not really depend on the temperature.

The Elastic Network movements are mostly stretching of the beta sheets or rotations around the center of the protein which are clearly visible. The movements of the elastic network are much stronger than these of Gromacs.

By summing up the movements of the Gromacs models and the elastic network models are somehow similar. Because of this fact, we think the modeling of the movement is correct and it could be the real vibration of 1BPT.

NMA vs. MD analysis

Normal modes is more a slight insight into the movements of a protein. Because it uses the harmonic approximation approach, all results have some attraction and repulsion mostly in big parts of the protein. But it is very clearly, that these movements are not the natural movements.

Molecular Dynamics analysis shows more the detailed parts of protein movements. It goes deeper into details like twisting side-chains or total movement of the whole protein. It takes much more time to simulate these processes and all simulation are time-dependent. Therefore the simulation shows only some nanoseconds of a protein movement, mostly mere a blink into the proteins life.

References

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