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**|**PLoS Comput Biol**|**v.3(9); 2007 September**|**PMC1988854

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PLoS Comput Biol. 2007 September; 3(9): e172.

Published online 2007 September 21. doi: 10.1371/journal.pcbi.0030172

PMCID: PMC1988854

Michael Levitt, Editor^{}

Department of Computational Biology, School of Medicine, University of Pittsburgh, Pittsburgh, Pennsylvania, United States of America

Stanford University, United States of America

* To whom correspondence should be addressed. E-mail: bahar/at/ccbb.pitt.edu

Received 2007 April 2; Accepted 2007 July 20.

This article has been cited by other articles in PMC.

Elastic network (EN) models have been widely used in recent years for describing protein dynamics, based on the premise that the motions naturally accessible to native structures are relevant to biological function. We posit that equilibrium motions also determine communication mechanisms inherent to the network architecture. To this end, we explore the stochastics of a discrete-time, discrete-state Markov process of information transfer across the network of residues. We measure the communication abilities of residue pairs in terms of hit and commute times, i.e., the number of steps it takes on an average to send and receive signals. Functionally active residues are found to possess enhanced communication propensities, evidenced by their short hit times. Furthermore, secondary structural elements emerge as efficient mediators of communication. The present findings provide us with insights on the topological basis of communication in proteins and design principles for efficient signal transduction. While hit/commute times are information-theoretic concepts, a central contribution of this work is to rigorously show that they have physical origins directly relevant to the equilibrium fluctuations of residues predicted by EN models.

In recent years, there has been a surge in the number of studies using network models for understanding biomolecular systems dynamics. Essentially, two different groups of studies have been performed, driven by two different communities. The first is based on molecular biophysics and statistical mechanical concepts. Normal mode analyses using elastic network models lie in this group. The second is based on information theory and spectral graph methods. The present study demonstrates for the first time that signal transduction events directly depend on the fluctuation dynamics of the biomolecular systems, thus establishing the bridge between the (newly proposed) information-theoretic and the (well-established) physically inspired approaches. We have applied the new approach to five different enzymes. Functionally active residues are shown to possess enhanced communication propensities. Furthermore, secondary structural elements emerge as efficient mediators of communication. These results provide us with important insights for protein design and mechanisms of allostery.

Proteins function neither as static entities nor in isolation, under physiological conditions. They are instead subject to constant motions and interactions, both within and between molecules. These motions can be either random fluctuations or concerted functional changes in conformations; and their sizes can vary from localized motions (e.g., single amino acid side chain reorientations) to large-scale global motions (e.g., domain–domain or intersubunit movements). While motions in the nanoseconds regime can be explored by full atomic simulations, understanding those involving large-scale structural rearrangements remains a challenge. In recent years, elastic network (EN) models in conjunction with modal analysis, and in particular the Gaussian Network Model (GNM) [1–3], have been widely used for elucidating the collective dynamics of proteins and exploring their relevance to biological function [4–9].

We posit that these collective motions also determine communication patterns that are inherent to the native architecture. To explore the validity and implications of this concept, we assume a discrete-time, discrete-state Markov process [10–11] of “information” transfer across the network of residues and measure two basic quantities: *hitting time* and *commute time* [11]. Hitting time *H*(*j,i*) is the expected number of steps it takes to send information from residue *v _{i}* to residue

A major goal in this study is to relate the hitting (and commute) times derived from the Markovian stochastics model to the equilibrium fluctuations (mean-square fluctuations and cross-correlations) of residues predicted by EN models, thus bridging the gap between two disciplines, information theory and statistical mechanics. To this end, using the theory of generalized matrix inverses [12–14], we show that hitting/commute times can be expressed in terms of the Kirchhoff matrix of inter-residue contacts that underlie the GNM methodology. Additionally, we present new insights into the signal transduction properties of enzymes, the catalytic residues of which are shown to be distinguished by their fast and precise communication abilities.

The paper is organized as follows. The Results are divided into three parts: first we present the Markovian stochastic model of information diffusion developed for exploring the inter-residue communication in proteins. The process is controlled by transition probabilities for the passage/flow of information across the nodes, which in turn is based on the internode affinities derived from atom–atom contacts in the folded structures. Second, we describe the evaluation of hit and commute times, and illustrate these concepts by presenting the application of the methodology to five different enzymes. Strikingly, active residues are distinguished by their effective communication stochastics. Third, we present a rigorous derivation of the mathematical relation (Equation 15) between inter-residue hit/commute times, and their fluctuation dynamics derived from purely statistical mechanical theory. This important relation establishes the bridge between information-theoretic quantities evaluated here for proteins and the intrinsic structural dynamics of proteins as described by physics-based models, and provides a new avenue for further examination of protein allostery using the new information-theoretic perspective.

The protein structure is modeled as a network of *n* nodes, each representative of a given residue *v _{i}*, for 1 <

where *N _{ij}* is the total number of

The affinities provide a measure of the local interaction density *d _{j}* at each residue

and *γ* is a force constant uniform over all springs.** Γ** is also called the

A *discrete-time, discrete-state* Markov process [11,16] is defined by setting the communication probability between residue pairs to be a function of their affinity. In particular, we define

as the conditional probability of transmitting information to residue *v _{i}* in

The conditional probability matrix ** M** = {

Suppose the probability of initiating the Markov propagation process at node *j* is *p _{j}*(0). Then, the probability of reaching residue

where ** p**(

Assume there is a path connecting every pair of residues in the network. Then, as the number of steps *β* approaches infinity, ** p**(

Whereas the evolution of the diffusion process is a function of the starting distribution, the stationary distribution is invariant to the details of initiation.

In the continuous time limit [17], the change in probabilities follows the master equation
= (** M** −

The hitting time *H*(*j,i*) is the average number of steps it takes for the information residing at residue *v _{i}* to be transmitted to residue

The calculation of *H*(*j,i*) requires the consideration of all possible pathways on the network, each being weighted by the product of transition probabilities along the path, starting from *v _{i}* and ending at

where Equation 4 is used on the first term on the right hand side. By definition, *H*(*i,i*) *=* 0. Equation 8 provides a self-consistent method for evaluating the hitting time between any two nodes.

The commute time is defined by the sum of the hitting times in both directions, i.e.,

Note that the commute time is symmetric by definition while *H*(*i,j*) is not, as will be illustrated below for example proteins. See the section “Pedagogical example to compute hit/com-mute time” in Methods for hit time analysis of a simple network.

In the calculations below, it proves convenient to define the average hitting times in both directions, as well as the average commute time, for each individual residue as

<*H _{r}*(

Figure 1A displays the hitting times *H*(*j,i*) computed for all residue pairs (*v _{i,} v_{j}*) for an example enzyme, snake phospholipase A2 (Protein Data Bank (PDB) [19], 1bk9 [20]). The blue regions correspond to short hit times, and red regions to long hit times, as indicated by the scales on the right. The map consists of the elements of the hitting time matrix

The higher ability of particular residues to transduce signals is also reflected in the commute times displayed in Figure 1B. The commute time is symmetric by definition, but the hitting time is not. The blue regions along the diagonal show that there is efficient communication along sequential residues, although we also observe several sequentially *distant* residue pairs that efficiently communicate. While the majority of these residue pairs are spatially close, as will be shown below, there is not necessarily a one-to-one correspondence between commute times and spatial distances, and some residue pairs emerge as more efficient communicators than others despite their longer physical separation.

Figure 1C displays the mean hitting time <*H _{r}*(

It is of interest to examine the signal transduction properties of catalytic residues. Phospholipase A2 has three catalytic residues: His48, Tyr52, and Asp99. Notably, all three residues (indicated by blue dots) are found to be located in minima (Figure 1C), i.e., the effective time required for these residues to establish communication with others is minimal.

To additionally highlight the enhanced communication properties of the catalytic residues, we plot in Figure 2 the mean <*H _{r}*(

Figure 3 illustrates similar results for four other enzymes: HIV-1 protease [21], ricin [22], human rhinovirus 3C protease [23], and endo-1,4-xylanase [24] (see caption for more details). The catalytic residues (highlighted as red dots) exhibit relatively short and narrowly distributed hitting times in each case. Ligand-binding residues (blue dots), on the other hand, display a wider range of hitting times and deviations, consistent with the results for phospholipase A2. At least one of the catalytic residues, indicated by the label, is distinguished in each case by its high communication speed and precision.

Consider the hitting time to the *n*^{th} residue *v _{n}* starting from residue

Here,
denotes *n*^{th} row of the hitting time matrix ** H** truncated to the first

or, in component form,

As derived in Methods, can be expressed in terms of the pseudo-inverse using the theory of generalized matrix inverses [12–14], to obtain

for the hitting time from residue *v _{i}* to any arbitrary residue

The above equation constitutes the most important result from the present study: it provides the physical basis for the hitting times obtained with the information-theoretic methodology by relating them to correlations between residue fluctuations derived from statistical mechanical theory [25,26]. The meaning of Equation 15 will be further elaborated below upon assessment of the contribution of each term in brackets.

Substitution of Equation 14 in Equation 9 yields an expression for the commute time in terms of *Γ*^{−1},

which, using Equation 19, reduces to

This is our final expression bridging commute times with fluctuations
in inter-residue distances. Note that the term in parentheses is a constant for all pairs of residues. Thus, the commute time between residues *v _{i}* and

The hitting time expression Equation 14 involves three different types of contributions: a one-body term that depends on the destination node,
; a two-body term that depends on the initial and final nodes,
; and a series of three-body terms that depend on intermediate nodes, in addition to the two end points,
*d _{k}*. Of interest is to understand the relative contributions of these three terms. Note that the first is always positive, and increases with the size of destination residue fluctuations; the second may be positive or negative, and the negative sign in front of this term implies that positively correlated residue pairs shorten the hitting time. Likewise, the third term may be positive or negative.

Figure 4 shows the results for phospholipase A2. Figure 4A–4C corresponds to the respective one-body, two-body, and three-body contributions. Note that Figure 4A–4C has different scales, for clearer visualization. As we demonstrate in Figure 4A, the one-body term plays by far a dominant role in determining the resulting hitting times (shown in Figure 1A), i.e., the mean-square fluctuations of the destination node largely determine the hitting time. Residues subject to large amplitude fluctuations require a longer time to be hit, while those subject to small amplitude fluctuations, usually confined to the core or high-density regions, display short hitting times.

The two-body term may be positive or negative, depending on the type of cross-correlations between residues *v _{i}* and

The qualitative features observed here were verified to be valid for all examined proteins: mainly, the mean-square fluctuations of the destination node play a dominant role in determining the hitting (or commute) time, and the cross-correlations between the two end points may increase or decrease the hit/commute time, depending on the type of correlation. Anticorrelations have a retarding effect, while positive correlations reduce the hitting time. In the extreme case of the two nodes moving in phase, by the same amplitude, the effective hit/commute time approaches zero.

The commute times provide us with a means of estimating effective communication distances
between residues *v _{i}* and

for the mean-square distance traveled by a random walk of *n* steps, with *l* being the average step size. In our case, *l* can be readily estimated from the average distance between connected nodes in the network. For phospholipase A2, *l* is evaluated to be 3.41 Å. Note that this is shorter than the distance (~3.81 Å) between consecutive α-carbons, because side chain atoms between neighboring residues may get closer to each other. The number of steps, on the other hand, is directly given by the hitting times themselves (as hitting times are expressed in terms of number of steps). For simplicity, we will use *n =* ½*C*(*i,j*) for the effective number of steps for communication between residues *v _{i}* and

Figure 5A displays the results for phospholipase A2. The effective distance
(ordinate) is plotted therein against the physical distance
directly evaluated from the PDB coordinates, averaged out over all atoms of residues *v _{i}* and

Figure 5B and and5C5C displays from two different perspectives, two residues (Trp31 and Ile104) located at the same physical distance (11.35 ± 0.15 Å ) from His48, but differing in their communication distances (69Å and 46.5Å) by a factor of approximately 1.5, pointing to the importance of the particular topology, or secondary structural elements, in increasing the effectiveness of communication. The communication with Trp31, located on a loop, turns out to be much slower, in this case. Figure 5D and and5E5E illustrates the opposite case of two residues (Lys69 and Asn125) that display comparable communication distances (60.5 ± 1.52 Å), while their respective physical distances (12.9 and 21.6 Å) differ by a factor of 1.7, approximately.

The comparison of the effective and actual (physical) communication distances in Figure 5 suggests that secondary structural elements possess higher abilities in processing signals. To test the validity of this conjecture, we analyzed the distributions of hitting times *H*(*i,j*) to residue *i*, for the three cases where residue *i* is *α*-helical, *β*-strand, or coiled/disordered*.* Figure 6 displays the distributions obtained by combining the results for the examined enzymes. Because the average hitting time increases linearly with the size of a given enzyme, the results for each enzyme are normalized with respect to the number of residues N in each protein, before combining the data. *α*-helical residues are observed to be the most efficient communicators, succeeded by *β*-strand residues (intermediate behavior), while the coiled residues are slower and exhibit a broader distribution of hitting times. Examination of the individual proteins, on the other hand, reveals that *β*-strands may exhibit strong dependence on their spatial location in the 3-D structure of the proteins (Figure 7).

As noted above, the mean-square fluctuations of the destination node play a dominant role in determining the hitting (or commute) time. The higher communication propensity of *α*-helical, and to some extent *β*-strand residues may thus be rationalized by the smaller fluctuations of secondary structural elements compared with coiled regions commonly observed in proteins. It is worth noting, however, that these observations hold for single domain proteins. As illustrated in Figure 8 for a multidomain protein, adenylate kinase, the communication between residue pairs belonging to different domains are usually slower than that between pairs in the same domain.

Methods based on network models significantly helped in recent years in providing a comprehensible description of the dynamics of biomolecular systems. On the one hand, methods based on fundamental statistical mechanical principles have been proposed for delineating the collective motions of biomolecules [1–9]; on the other, those based on spectral graph theory and machine learning algorithms have been developed for exploring allosteric effects and response to perturbations/mutations in complex structures [29,30]. While these two methodologies concur in their objectives—understanding the complex machinery of biomolecular systems, the connection between these two approaches has been elusive due to their different originating disciplines, as well as the basic quantities they shed light onto: frequency spectrum and normal mode shapes in the former, shortest paths of communication, and hitting/commute times in the latter.

The present study offers a rigorous way of connecting the two approaches, by demonstrating that the commute times between residues *v _{i}* and

Notably, the application to example enzymes point to the more efficient communication propensity and precision of catalytic sites (Figures 1–3), to the role of residue fluctuations and their correlations in transmitting information (e.g., delaying effect of anticorrelated pairs), to the structure-encoded differences in the communication abilities of residue pairs, irrespective of their physical distances; and to the importance of both tertiary contact topology and local (secondary) structure in defining effective means of communication (Figures 5 and and6).6). Also, irrespective of physical distance, interdomain communication tends to be slower than intradomain, as illustrated in Figure 8 for adenylate kinase.

The major advantage of the present stochastic model over the GNM is the fact that the new methodology lends itself to a comprehensive assessment of the communication paths and their efficiency in biomolecular structures. As such, it holds promise for identifying allosteric communication pathways as well as the sites distinguished by high allosteric potentials, thus providing insights into the design principles of biomolecular machines. The presently observed enhancement in the information transfer properties of catalytic residues and secondary structural elements suggests possible design requirements for efficient enzymatic activity. In this context, it is worth noting the relevant studies by Choe and Sun [34] and Maritan and coworkers [35], which point to the dependence of equilibrium dynamics on secondary structural content/type. It remains to be understood whether such special communication abilities of catalytic residues result from their local packing topology or more global features conferred by evolutionary pressure.

We note that finding suitable experimental setup for probing hit-times is a challenge. In general, the residues/interactions involved in information flow, or the changes in inter-residue distances (which directly define the commute times) may be assessed by site-directed mutagenesis and cross-linking experiments as well as spectroscopic methods such as site-directed fluorescence labeling [36] or FRET [37].

Finally, establishing the bridge between these two disciplines will permit us to translate the wealth of concepts and methods developed in information-theoretic approaches, to exploring the signal transduction mechanisms in complex biomolecular systems, thus complementing physically inspired models and methods.

Let Δ*r** _{i}* and Δ

where *k _{B}* is the Boltzmann constant and
denotes the

The mean-square fluctuations
in inter-residue distances Δ*r** _{ij}* = Δ

Equation 20 can be rewritten, using Equation 19, in terms of the elements of *Γ*^{−1} as

By definition, ** Γ** is positive semi-definite, i.e.,

Consider a small undirected network of three nodes connected as in

Assume that *a _{ij}* =

Similarly,

Simultaneous solution of Equations 22 and 23 yields *H*(*k,i*) *=* 4 and *H*(*k, j*) *=* 3. The second way to unroll the recursion is to enumerate the paths between pairs of nodes, as shown in Table 1. The enumeration leads to the calculation of expected time steps

Given the symmetry in the network here, the hitting time *H*(*k,i*) *= H*(*i,k*)*,* but this may not be true in general. To conclude this example, consider the hitting time from *j* to *i*. Again, the random walk unrolled partially is shown in Table 2. This enumeration leads to

Clearly, the hitting time *H*(*j*,*i*) ≠ *H*(*i*,*j*). While iterative methods, using Equation 8, are one way to solve for hit/commute times, there is also a “fundamental matrix” technique [10] for computing these quantities.

The discussion below borrows from results in [12,13]. Deriving
from *Γ*^{−1} is a three-step process: (i) put together a matrix
(size: *n* − 1 × *n*) from
(size: *n* − 1 × *n* − 1) by appending a column vector ** p** (size:

(ii) derive ** Γ** (size:

(iii) following the theory of generalized inverses [12] use
to express the inverse
and then derive
from *Γ*^{−1}.

The vectors ** p** and

and

which implies that

The generalized matrix inverse of is given by

where

and

Substituting for ** p** we obtain
, and hence

is a rank-1 update to
and *r** ^{T}* is a row vector. So, we can tease out
here by

such that in component form

Now when we write

the effect of adding a row vector *t** ^{T}* to
is similar to adding the column vector

Putting the inverses in Equations 38 and 39 together, we obtain

By substituting Equation 40 into Equation 13, we get

Using symmetry, the summation can be extended to *n* as in

This derivation of hitting time to *n*^{th} residue *v _{n}* from

We gratefully acknowledge the contribution of Sinem Ozel to the results on adenylate kinase.

- EN
- elastic network
- GNM
- Gaussian Network Model

**Author contributions.** CC and IB conceived and designed the experiments, performed the experiments, analyzed the data, and wrote the paper.

**Funding.** This study was supported by US National Institutes of Health grant R33 GM068400–01.

**Competing interests.** The authors have declared that no competing interests exist.

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