**Theory**

** **

**Hitting/Commute Time
– Intramolecular Communication**

The Markovian
stochastic model of information diffusion has been developed for exploring the
inter-residue 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 *i*-*j* 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

**G **=
**D**** **-** A**** ** (4)

Based on this, the hitting time (information-theoretic quantities) was bridged to the GNM-defined intrinsic structural dynamics (statistical mechanical theory) of proteins.

Residues *v _{i}*
and

** **(5)

Then we are
able to calculate the hitting time between two any nodes from the above
equation using a self-consistent method. The commute time between *v _{i}*
and

*C***( j,i)= H(j,i)
+ H(i,j) = C(i,j)**

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 pseudo-inversion of

** ** (8)

**Figure
1**. Hitting time (A)
of Phospholipase A2 (PDB id: 1BK9) and the decomposed hitting time: one-body
term (B), two-body (C) term and three-body 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: one-body term, which
is the mean-square fluctuation of the response site (<*∆r _{j}^{T}∆r_{j}*>);
two-body term, which depends on the cross-correlations between residues

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.