**Theory**

** **

**Gaussian Network Model (GNM)**

** **

**Figure 1**: Schematic representation of equilibrium
position vectors of residues, their distance vectors and fluctuations.

The**
GNM **is** **based on the statistical mechanical theory developed by Flory
and coworkers for polymer gels (1),
where network junctions undergo Gaussian fluctuations. In the GNM, the
positions of the network junctions/nodes are identified with the C^{a}
atoms, and the elastic springs represent the interactions that stabilize the
native contacts. Two residues are assumed to be in contact when their a-carbons
are separated by less than a cutoff distance *R _{c}*, which is
usually taken around 7 Å. As shown schematically in

Δ R - _{ij}R_{ij}^{0}^{ }= ΔR– _{j }ΔR_{i} |
(1) |

where
*R** _{ij}* is the instantaneous distance
vector,

(2) |

Here
**G** is the **Kirchhoff matrix**, the off-diagonal
elements of which are defined as

Γ R≤ _{ij }R_{c}Γ R> _{ij }R _{c} |
(3) |

and the diagonal elements are

Γ _{ij} |
(4) |

where
the summation is performed over all off-diagonal elements in the *i ^{th}*
row (or column);

The
**cross-correlations** between residue fluctuations are found from the
statistical mechanical average

* *

(5) |

where Γ^{-1}* _{ij}* is the

The determinant of **Γ** is 0, and hence **Γ**^{-1
}cannot be
calculated directly. Instead, it is found from the eigenvalue decomposition **Γ** = **U****L****U**^{T} which results in an ensemble of *N*-1 independent
modes. **U** is the orthogonal matrix whose *k ^{th}* column

The cross-correlation between
residues can be written as a sum of *N*-1 GNM modes as

(6) |

The **mean-square fluctuations**
of residue *i* in mode *k* can be evaluated from Eq. 6 by replacing *j*
by *i*. The plot of *tr*[*u*_{k }**u**_{k}^{T}**]
**as a function of residue *i * represents the probability distribution
of residue square fluctuations in mode *k*, also called *k ^{th}
mode profile. *

* *

The cross-correlations are usually
shown in a matrix/map format, with the diagonal terms representing the
mean-square fluctuations. This *ij*^{th} element of this matrix,
termed cross-correlation matrix, is

ΔR_{i }. ΔR>_{j } |
(7) |

The normalized cross-correlations are given by

** **

ΔR_{i }. ΔR> / [< _{j }ΔR_{i }. ΔR> < _{i }ΔR_{j }. ΔR>]_{j }^{1/2} |
(8) |

**C**_{ij }^{(n)}* _{ }*varies in the range [-1, 1] and provides
information on

**Degree of
collectivity**

** **

The degree of collectivity of a given mode measures the extent to which the structural elements move together in that particular mode. A high degree of collectivity means a highly cooperative mode, that engages a large portion of (if not the entire) structure. Conversely, low collectivity refers to modes that affect small/local regions only. Modes of high degree of collectivity are generally of interest as functionally relevant modes. These are usually found at the low frequency end of the mode spectrum.

Collectivity for a given mode *k*
is a measure of the degree of cooperativity (between residues) in that mode,
defined as (8)

(9) |

where,
*k* is the mode number, and *i* is the residue index.

**Reference**

1. Flory,P. (1976) Statistical
thermodynamics of random networks. *Proc. R. Soc. Lond. A*, **351**,
351-380.

2. Bahar,I., Atilgan,A.R. and
Erman,B. (1997) Direct evaluation of thermal fluctuations in proteins using a
single-parameter harmonic potential. *Fold. Des.*, **2**, 173-181.

3. Bahar,I., Atilgan,A.R.,
Demirel,M.C. and Erman,B. (1998) Vibrational Dynamics of Folded Proteins:
Significance of Slow and Fast Motions in Relation to Function and Stability. *Phys.
Rev. Lett.*, **80**, 23.

4. Bahar,I. and Rader,A.J. (2005)
Coarse-grained normal mode analysis in structural biology. *Curr. Opin.
Struct. Biol.*, **15**, 586-592.

5. Rader,A.J., Chennubhotla,C.,
Yang,L.W. and Bahar,I. (2006) The Gaussian network model: Theory and
applications. In Bahar,I. and Cui,Q. (eds.), *NORMAL MODE ANALYSIS: THEORY
AND APPLICATIONS TO BIOLOGICAL AND CHEMICAL SYSTEMS*. Chapman &
Hall/CRC: Boca Raton, FL, pp. 41-64.

6. Eyal,E., Dutta,A. and Bahar,I.
(2011) Cooperative dynamics of proteins unraveled by network models. *WIREs
Comput Mol Sci*, **1**, 426-439.

7. Yang,L.W. (2011) Models with
energy penalty on interresidue rotation address insufficiencies of conventional
elastic network models. *Biophys. J.*, **100**, 1784-1793.

8. Brüschweiler,R.
(1995) Collective protein dynamics and nuclear spin relaxation. *J. Chem.
Phys.*, **102**, 3396-3403.