Home 
DynOmics 1.0 
Tutorials 
Theory 
References 
iGNM 2.0 
Mol. Sizer&Timer 
ANM 2.0 
Comp. Biol. Lab 
PITT site

Theory
Hitting/Commute Time – Intramolecular Communication
The Markovian stochastic model of information diffusion has been developed for exploring the interresidue communication in proteins (1). The first passage time in Markovian process, which is the average time (number of steps) for residue/node i to transfer the “message” from node i to j for the first time, is defined as hitting time H(j,i) and H(i,i) = 0. The process is controlled by transition probabilities for the passage of information across the nodes (residues). Specifically, the atomic contact affinity m_{ij} between pair of nodes ij defines the conditional probability which consist of the transition matrix M = {m_{ij}}.
(1)
where
(2)
and
(3)
A = {a_{ij}} and D = diag{d_{ij}} are the affinity and degree matrices respectively. N_{ii} is the number of heavy atom contacts between residues i and j based on a cutoff distance of r_{c} = 4 Å. N_{i} and N_{j} are the number of heavy atoms in residue i and j respectively. Notably, the Kirchhoff matrix (combinatorial Laplacian) G in GNM theory can be defined as
G = D  A (4)
Based on this, the hitting time (informationtheoretic quantities) was bridged to the GNMdefined intrinsic structural dynamics (statistical mechanical theory) of proteins.
Residues v_{i} and v_{j} are termed as the broadcaster (perturbation site) and receiver (response site) respectively. The passage from v_{i} to v_{j} can be performed in two stages: (i) from v_{i} to its directly connected neighbor residues (termed as intermediate residue v_{k}) that is one step away (∑^{n}_{k}_{=1 }m_{ki} = 1); (ii) succeeded by probabilistic passages from v_{k} to the final destination v_{j}. An efficient recursive formula can be derived as
(5)
Then we are able to calculate the hitting time between two any nodes from the above equation using a selfconsistent method. The commute time between v_{i} and v_{j} is termed as
C(j,i)= H(j,i) + H(i,j) = C(i,j) (6)
The commute time matrix C = {C(j,i)} is symmetric while the hitting time matric H = H(j,i) is not.
Technically, the “fundamental matrix” technique can also be used to calculate these quantities (2).
Based on the bridging between G and A in Eq. 4, we obtained
(7)
where G^{1} is the pseudoinversion of G. The elements in G^{1} are to the intrinsic dynamics of residues in GNM theory. We can rewrite the Eq. 7 as
(8)
Figure 1. Hitting time (A) of Phospholipase A2 (PDB id: 1BK9) and the decomposed hitting time: onebody term (B), twobody (C) term and threebody term (D). The figures were reproduced from (1).
The detailed derivation of Eq. 7 can be seen in (1). Based on the understanding of this equation, we can tell that the hitting time can be decomposed into three terms: onebody term, which is the meansquare fluctuation of the response site (<∆r_{j}^{T}∆r_{j}>); twobody term, which depends on the crosscorrelations between residues v_{i} and v_{j} (<∆r_{j}^{T}∆r_{i}>); and the threebody term, which depends on the crosscorrelations between intermediate residue v_{k} and residues v_{i} and v_{j} (<∆r_{k}^{T}∆r_{i}>  <∆r_{k}^{T}∆r_{j}>). Take the protein Phospholipase A2 (PDB id: 1BK9) as an example, the hitting time (Fig.1 A) of protein Phospholipase A2 was decomposed into three terms (Fig.1 BD). The onebody term (Fig.1 B) plays a dominant role the twobody term (Fig.1 C) has the visible effect in specific regions. While the effect of the three body term (Fig.1 D) is negligibly small.
In the example of Phospholipase A2, the average Receiver (response site) hitting times of the three catalytic residues H49, Y52 and D99 are in a smallest level (sensitive; efficient to receive signals).
Reference
1. Chennubhotla, C. and Bahar, I. (2007) Signal Propagation in Proteins and Relation to Equilibrium Fluctuations. PLoS Comp. Biol., 3, e172.
2. Norris, J.R. (1997) Markov chains. Cambridge (United Kingdom): Cambridge University Press.