BACKGROUND
cooperative protein response and thus cooperative network behavior requires information transfer between distal sites in a protein or protein complex.
protein structures often achieve such long range communication by allosteric movement  <cit> , but this is certainly not a requirement.
essentially any change in the dynamic properties of protein residues, upon ligand binding for example, that efficiently propagates through the structure and is detectable at a distal site, constitutes a form of signal transduction  <cit> .
many theoretical and experimental studies have been devoted to the structural dynamics of globular proteins and its implications for protein function.
recent experimental studies, on the one hand, focus on the role of protein dynamics for catalysis  <cit> , signal transduction  <cit> , cooperative response  <cit>  and protein aggregation  <cit> .
on the other hand, theoretical approaches have probed the relationship between protein structural dynamics and signal transduction, limited only by the combinatorial complexity associated with the conformational space.
in response, techniques such as targeted molecular dynamics  <cit> , anisotropic thermal diffusion  <cit>  or go-like sampling  <cit>  were developed to reduce the degrees of freedom so as to allow the mapping of a pathway that connects one ground state to its allosteric counterpart.
additionally, ranganathan and co-workers have mapped residues that participate in signal transduction in several important proteins by extracting evolutionary correlated mutations from multiple sequence alignments  <cit> .
together these studies show that  changes in protein dynamics can propagate through the protein structure thereby creating long range correlations between distal active sites  <cit> ,  only a fraction of residues in a protein structure participate in signal propagation  <cit> , and  these intra-protein communication modes are generally conserved within protein families and even protein folds  <cit> .
now that the principles relating correlated structural dynamics to signal transduction mechanisms within proteins are becoming apparent, we here present a method to identify and quantify these structural fluctuations in terms of information allowing computation of the information transfer between active sites.
in particular, we use information theory  <cit> , and more specifically the concept of mutual information, as a means to describe the relationship between structural protein dynamics and signaling behavior.
this association directly follows from the relation between the definitions of entropy as a measure of structural disorder in statistical thermodynamics, on the one hand, and as a measure of error on communication channels in information theory on the other hand  <cit> .
the potential of globular proteins to convey information throughout their structure, thereby correlating the behavior of distant effector sites, is indeed provided by the change in structural dynamics induced by ligand binding.
but more fundamentally, information transfer originates from mutual conformational dependence of the different residues composing a protein or protein complex.
in other words, when the conformational flexibility of one residue affects the conformational flexibility of another residue, changes in the structural dynamics will produce correlated changes in entropy and information is thus exchanged  <cit> .
mutual information quantifies the amount of conformational dependence between protein residues.
as a result a protein can be considered as a network of residues exchanging conformational information.
this interpretation provides a theoretical framework to understand how the redistribution of thermodynamic perturbations within a network of protein residues constitutes an information transfer network.
to illustrate our information theoretic approach, we map the residue-based information transfer network of the sh <dig> domain of the fyn tyrosine kinase  <cit> , a member of the src family of kinases  <cit> , that are involved in essential signaling pathways, for example controlling cell growth, proliferation, differentiation, motility and cell adhesion  <cit> .
family members share a common multi-domain architecture, with the archetypal src homology  <dig>  domain at the n-terminal  <cit> , a src homology  <dig>  domain  <cit> , the tyrosine kinase domain and a short c-terminal tail.
in its inactive form the kinase domain is sequestered by the sh <dig> and sh <dig> domains .
the sh <dig> domain docks to the linker connecting the sh <dig> to the kinase domain.
the sh <dig> domain holds on to a c-terminal phosphotyrosine sequence .
dephosphorylation of this c-terminal sequence leads to its release from the sh <dig> phosphopeptide-binding site.
this in turn leads to the release of the sh2-kinase linker from the sh <dig> domain and undocking of the kinase domain which is more than  <dig> angstrom  away from the sh <dig> phosphopeptide binding site  <cit> .
the sh <dig> domain has a typical length of approximately  <dig> amino acids and displays a simple fold  in which a central anti-parallel β-sheet is flanked by two alpha-helices.
the β-strands are commonly annotated from βa to βg, whereas the α-helices are referred to as αa and αb.
panel b  of figure  <dig> shows the labeled structure  of the fyn sh <dig> domain that is used in the present analysis .
the peptide binding site of the fyn sh <dig> domain consists of two functional regions: the phosphotyrosine binding pocket , responsible for ptyr recognition, and the hydrophobic binding pocket, which interacts with the residues c-terminal to the ptyr   <cit> .
the first binding cavity is located between the central anti-parallel β-sheet and the α-helix αa.
the second binding pocket is located on the opposite side of the phosphotyrosine binding cavity, between the central β-sheet and the α-helix αb.
in figure 2c, the names of the residues involved in binding the c-terminal phosphorylated sequences are annotated onto the structure following the nomenclature defined in  <cit> .
these names, e.g.
hisβd <dig>  will be used when referring to residues of either binding cavity.
other residues, e.g.
ser <dig>  will be indicated by their amino acid name and position in the fyn sh <dig> pdb structures used here .
for further detailed discussion of sh <dig> function and nomenclature we refer to the literature  <cit> .
experimental work on the role of the linker between the sh <dig> and sh <dig> domain of fyn has shown that the nature of the linker, i.e.
the residue pattern, determines both orientation and coupling between the two domains  <cit> .
as such it has a direct effect on the activation behavior of the protein kinase and on the intra-molecular communication.
this observation is supported by experimental and molecular dynamics work performed by kuriyan et al.
<cit>  on the src kinase.
they showed that the release of the c-terminal sequence from the sh <dig> binding site uncouples the motions of the sh <dig> and sh <dig> domains by increasing the flexibility of the linker loop connecting these two domains.
the connector link between the sh <dig> and sh <dig> domain therefore acts as an inducible snap lock mechanism and replacement of three of the eight residues in this linker by glycines generates a constitutively active kinase.
in effect, the phosphopeptide binding site at one end of the sh <dig> domain is thus coupled to the sh3-sh <dig> linker at the opposite side of the sh <dig> domain.
importantly, cooperativity is achieved without major allosteric movement within the sh <dig> domain and therefore originates from subtle differences in dynamics between phosphopeptide bound and unbound states.
our analysis shows that mutual information identifies and quantifies the cooperativity in the sh <dig> domain over long distances, confirming other results for different non-allosteric structures  <cit> .
in other words, binding the phosphotyrosine peptide triggers an information exchange from the sh <dig> binding pockets toward residues located at the opposite side of the domain, targeting the linker regions.
hence, the sh <dig> peptide binding site acts as a switch that distributes information on its binding state to both linkers with the other domains, coordinating the sh2-sh <dig> docking module.
RESULTS
how to quantify the information exchange within a protein?
in shannon's theory of information transduction along noisy channels, where the level of noise corresponds to the fraction of the input characters that may be wrongly translated to output characters, information is quantified as the degree of certainty obtainable about the output signal, once a particular input is given  <cit> .
a random channel that produces for any input signal a random output has a maximum degree of uncertainty, whereas for a noiseless channel, which produces an exact copy of the input at the output, there is no uncertainty about the output.
this variation in the degree of uncertainty is expressed in terms of entropy, allowing for the immediate application of information theoretical principles to protein dynamics simulations in a straightforward manner  <cit> .
specifically, we focus here on the notion of mutual information to express the mutual dependence between the conformational states of residues in a protein, as a quantification of the information flow within.
for more information on information theory we refer to  <cit> .
as a tool, mutual information, and other co-variation measures, has been applied to identify co-varying or co-evolving residues in multi-sequence alignments  <cit> .
yet, as far as we are aware, no work exists that applied this informational theoretical measure to determine the conformation coupling between residue sidechains.
so when do two sidechains exchange information?
two linked residues are dependent when knowing one conformation will convey information about the conformation of the other.
in this case, the two residues share information.
for instance, consider two amino acid residues in a protein, each having some conformational variation in their side chains.
the mutual information shared by these residues is defined by the entropy reduction that is observed at the second residue when the conformation of the first is fixed in an arbitrary configuration .
thus, when ligand binding conformationally restricts the first residue then mutual information will quantify how much of the signal of ligand binding is received at the second residue since its conformational flexibility will also be restricted.
mutual information between two residues is thus zero when the conformational state of one residue does not provide any information about the conformational state of the other residue.
on the other hand, mutual information is maximal when each conformational state of a residue's sidechain uniquely defines the conformational state of the other residue's sidechain.
in accordance with information theory, the conformational space of each residue corresponds to the residue's alphabet.
exploration of the conformational space of the amino acids
the calculation of mutual information between any two amino acids in a protein requires reliable statistics of the conformational dynamics of all amino acids in the native structure of the protein .
the method relies on computing both the probability of residue i to adopt an arbitrary conformation independently and the joint probability of finding residue i and residue j in an arbitray combined combination.
this is achieved via effectively sampling the conformational states of each amino acid in a protein structure so that all possible combinations of conformations of residues i and j can be explored.
it is clear that the computational size of such a sampling problem can only be addressed by reducing the number of possible conformations that each amino acid can adopt to a relatively small number of discrete states.
to achieve this, we treat the sidechain and backbone flexibility of each amino acid as separate components of its overall conformational dynamics and consider only a finite number of states for each.
in order to keep the number of sidechain conformations computationally tractable, it is common to employ a discrete, finite-size alphabet for each residue, called rotamer library.
such rotamer libraries are constructed by extracting from a database of high resolution protein structures the most frequently occurring states and thus the degree of coarse graining that is employed to record the statistics imposes a resolution on the data.
for the purpose of detecting conformational dependencies between neighboring residues in a protein structure, we require a finer resolution than is provided by common rotamer libraries used for homology modeling and we thus constructed a database of backbone dependent sidechain conformations with a  <dig> degree resolution on the sidechain dihedral  angles .
given the backbone dihedral angles of a certain residue in the protein structure, a list of possible sidechain conformations and their probabilities can be retrieved from this database.
the number of chi angles in the residue determines the size of this list, meaning that residues with small sidechains tend to have shorter lists than residues with long sidechains.
the conformational state of the sidechain in the protein structure serves as a starting point for generating these lists.
information concerning the flexibility of the backbone of a protein can be obtained from either molecular dynamics simulations  or nmr data.
exploration of the conformational space accessible to the protein backbone using md is a computationally expensive method.
moreover, vendruscolo and co-workers have recently shown that md simulations yield more realistic results when restricted by experimental data in the form of residue-residue distances derived from nuclear overhauser effects  in a nuclear magnetic resonance  experiment  <cit> , suggesting that unrestricted md simulations are not the most effective method to sample the backbone ensemble.
the work of vendrusculo and others also revealed that the backbone dynamics can essentially be captured by a small number of representative backbone structures.
moreover, bahar and colleagues recently showed  <cit>  that nmr models can also be viewed as an ensemble of conformations accessible under physiological conditions and that even though the rmsd values reflect the uncertainties in the coordinates, they also contain physically meaningful contributions of equilibrium fluctuations  <cit> .
although this point is still under debate, we here found that sampling the sidechain conformations of each amino acid on the collection of backbones present in the nmr dataset using a monte carlo approach  yields results that are consistent with experimental studies as well as more exhaustive simulations performed by kuriyan et al  <cit> .
more specifically, we employed a metropolis algorithm implemented in the foldx force field  <cit>  from which an equilibrium distribution of sidechain conformations, compatible with a given protein backbone structure, is obtained .
the force field scores the conformations taking into account the packing interactions.
the sidechain sampling results for a single backbone structure obviously introduce a strong bias in the apparent information flux towards residues whose sidechains arbitrarily happen to be strongly coupled in this particular backbone conformation.
when information is obtained from an ensemble of related backbone conformations however, each backbone introduces slight variations in the pattern of residue-residue couplings.
as a result, the combination of the sidechain sampling on the entire ensemble of backbone structures acts as a filter that removes sporadic couplings while accumulating consistent couplings, thereby revealing the true network of information exchange between all residues.
we have here taken the entire ensemble of backbone structures in the nmr datasets of the fyn kinase sh <dig> domain as an adequate sample and have not systematically explored if a reduced number of backbones could be employed to the same effect.
construction of the residue-residue information network in fyn sh2
since the monte carlo sampling directly yields the equilibrium distribution of sidechain conformations observed at each position along the protein backbone, the quantity of mutual information between all residue pairs in the protein can be calculated using probability and entropy calculations .
in this way, information transfer over both short and long distances in the folded protein is elucidated, explicitly revealing the communication between all residues in the protein.
extracting only the changes in mutual information that result from the difference between bound and unbound states of the sh <dig> domain , highlights the change in communication patterns.
hence, those amino acids that experience the strongest changes in mutual information can be detected and mapped on the network of sidechain interactions in the protein, thereby revealing how series of dynamically coupled residues in the topology of the protein direct the overall communication.
in order to obtain the network of residue-residue couplings in the fyn sh <dig> domain, we employed a foldx based sidechain sampling on the backbone structures of this sh <dig> domain as determined by nmr on the protein domain in isolation and bound to its phosphopeptide ligand .
the nmr ensemble was determined using over 90% of the structurally non-redundant nuclear overhauser effects , so that the ensemble can be considered to possess adequate precision and accuracy  <cit> .
in order to ensure reliable statistics, we collected over  <dig> thousand samples of sidechain configurations from approximately  <dig> million simulation steps over the  <dig> backbone models present in the nmr ensemble, producing the probabilities of finding the residues' sidechain in a particular conformation.
from all these probabilities mutual information between all residue pairs was derived.
to capture how the information exchange within the sh <dig> domain changes due to ligand binding we require nmr structural data on both the bound and unbound state of the protein.
since there is no structural data on the unbound state of fyn sh <dig> available publically, we assume here that the backbone flexibility for bound and unbound state can be derived from the ensemble stored in the 1aou dataset by energy minimization .
this simplification has as result that large conformational changes in the backbone of the sh <dig> domain are not taken into account.
yet since domains do normally not experience large structural changes, this simplification may still provide viable results.
changes in mutual information link residues at long distances
to visualise the mechanistic implications for our model system a clustering algorithm  was applied on the mutual information matrix in figure 2a and mapped on the structure of fyn sh <dig>  figure 2c displays the residue cluster for which the mutual information increases most upon peptide binding .
only those pairs of residue whose mutual dependence is above this threshold are accepted in the cluster, meaning that they form a complete graph where the weight of each link is above the threshold value.
on the one hand, and as expected, these are composed of residues directly interacting with the phosphopeptide itself  but interestingly also consists of residues on the other side of the domain from which the sh2-sh <dig> and sh2-kinase linkers emerge .
even though previous studies of the fyn sh <dig> domain  <cit>  did not assign an important role to the conserved hisβd <dig>  our analysis suggests this residue experiences a very strong change in mutual information upon ligand binding with all other residues in the structure .
this strong effect may be the result of the displacement of the helix αa in this particular sh <dig> domain, making the ptyr-binding cavity narrower when the peptide is bound  <cit> .
most affected seems to be residue thref <dig>
thus, our analysis reveals a strong coupling between the peptide binding site of fyn sh <dig> and its sh3-sh <dig> and sh2-kinase linkers.
these first results show that the sampling approach discussed here provides a direct way to identify and quantify highly coupled groups of residues.
moreover, it confirms the notion that subtle changes in structural dynamics can effectively couple residues at distal locations in the structure.
note that the clustering does not show how this information exchange actually occurs, i.e.
it does not provide a causal explanation.
it provides an identification of the residues that may be involved with signaling.
yet, the idea is that when, by lowering the threshold, a coherent collection is obtained, possibly identifying a consecutive path that links the binding region to other parts of the domain structure.
closer inspection of the information exchange for some residues
to understand the key elements of information exchange in proteins we compared the change in mutual information of both silent and informative residues with the rest of the protein.
we will here illustrate our analysis by two examples that recapitulate the key features of both informative as well as silent residues.
ser <dig> will serve as an instance of the collection of silent residues whereas hisβd <dig> represents a prominent member of the informative residues.
specifically, we mapped the change in mutual information of these residues with all the other residues of the sh <dig> domain onto the structure.
strong changes  are colored red, weak changes  blue and intermediate changes are white.
in figure 3a, we observe for hisβd <dig> that three classes of residues contribute significantly to its score.
first, locally interacting residues directly influence each other .
second, the phosphorylated peptide connects all the relevant residues in the binding pocket, creating a channel by which conformational fluctuations can be transmitted .
as a consequence, these residues are mutually dependent on the state of hisβd <dig>  third, hisβd <dig> is also coupled to a number of residues at the other side of the protein.
we see in figure 3a  that the residues at the kinase linker  that were previously detected by clustering are again present, i.e.
long-range communication between the histidine and residues val <dig>  pro <dig> and val <dig> is clearly visible here.
moreover we see also a coupling with a number of residues related to the sh3-sh2-linker region , meaning that information is exchanged with both linker regions when switching between unbound and bound state.
the results for ser <dig> are strikingly different.
in figure 3b, we see that when the phosphorylated peptide becomes attached to the sh <dig> domain, only a few residues located in the peptide binding site exchange more information with this particular residue.
further, informative and silent residues do not only strongly differ in terms of the amount of residues with whom they are coupled but also in terms of the distances over which they communicate.
figure  <dig> shows that whereas ser <dig> is conformationally isolated, hisβd <dig> has an extensive network of couplings with both proximate as well as distal residues.
contiguous communication pathway in fyn sh2
as argued in previous sections, our information theoretical approach provides a method to quantify the change in exchange of mutual information between all pairs of residues of the fyn sh <dig> domain as a consequence of ligand binding.
our analysis revealed that for the majority of residues the conformational coupling to the rest of the protein is not altered upon peptide binding and as a consequence they can be considered silent in terms of signal transduction.
a small fraction of residues, however, experience a significant change in conformational coupling and are thus information-rich.
clustering only the most informative of these residues  revealed a strong coupling between the peptide binding site and the region harboring the linkers connecting the fyn sh <dig> to the other domains of the fyn kinase.
it remains of course to be explained by which structural mechanism these distal sites become coupled.
as argued earlier, the current framework can only identify and quantify the residues that are involved in the interaction.
to identify the causal relationships a different analysis is required.
several mechanisms for signal transduction have been described in the literature  <cit>  and as such our method does not provide the means to unequivocally single out a given mechanism.
however, clustering the informative residues and mapping these on the structure of the protein can provide substantial information.
for instance the relative occurrence of coupling versus uncoupling pairs and their disposition on the topology of a protein domain could allow to distinguish pathway models  <cit>  from allosteric models  <cit> .
as our model only displays increase in conformational coupling upon ligand binding, suggesting a pathway model, we here extract the group of residues that define the pathway through clustering  of informative residues that have a mutual affinity higher than a particular noise level .
in figure  <dig> we see how the pathway is formed by decreasing the clustering threshold from  <dig>  to  <dig>  bits: a contiguous dynamic pathway  <cit>  emerges, involving the previously identified residues in the peptide binding site and the linker region.
as discussed above only a subset of residues of the sh <dig> domain is involved in signal transduction, whereas the other residues seem to be conformationally isolated from ligand binding.
importantly, signal transduction is achieved throughout the core of the sh <dig> domain and involves the main secondary structure elements of the structure.
hence, we show that the fyn sh <dig> domain acts as indivisible information transmission unit propagating information from its binding site over the tertiary structures to the linker regions of the domain.
CONCLUSIONS
driven by the availability of genomics and proteomics data, biological research is increasingly focused on cellular networks rather than on individual proteins.
accordingly, a major effort in systems biology is devoted to understanding how topology shapes macromolecular networks into functional units performing information processing tasks such as logical gating, band-pass filtering or signal modulation.
biological networks are generally represented by graphs of interconnected protein nodes.
in reality, however, these networks form macromolecular complexes whose assembly is determined by intermolecular atomic interactions.
as shown by several experimental and theoretical studies, ligand binding may induce changes in the structural dynamics of proteins.
these perturbations propagate cooperatively throughout the protein, coupling distal locations and thereby effectively transferring information.
hence, proteins domains are the elementary units of information processing in cellular organisms.
as such proteins can themselves be represented as networks of interconnected residues exchanging information.
protein complexes can by extension be modeled in a similar way.
the method introduced here allows to quantitatively translate protein structural dynamics in terms of shannon's information theory.
moreover, mutual information permits to determine how much information is exchanged between each pair of residues within a protein structure.
transposing structural dynamics in terms of shannon information has the advantage of providing a description level that matches the relevant functional features – i.e.biological information,- that we aim to extract from signal transduction pathways.
moreover, the analysis does not make any assumptions on the information transmission model.
it strictly focuses on identifying the residues whose coupling increases or decreases as a consequence of peptide binding.
further, the method is applicable in identical manner to residue networks at atomic scale as well as to macroscopic descriptions of cellular networks.
as explained in results, the mutual information exchanged between residues in protein residue networks is simply a statistical thermodynamics description of the conformational coupling between these residues.
by discretising conformational space at the residue level, we easily find the average shannon information for each residue by calculating the average entropy of the conformational probability distribution for each residue.
the mutual information exchanged between two residues is the difference of the shannon entropy from one residue and the conditional shannon entropy of that residue given knowledge of the probability distribution of the other residue.
in other words, it represents the amount of coupling between two protein residues.
once the mutual information is calculated for each pair of residues, clustering can be applied to determine the structural mechanism by which information is transferred between functional sites.
in effect we hereby consider a protein as an instance of a noisy coding channel linking a probability distribution of input signals at one binding site with a probability distribution of output signals at another binding site.
for the fyn sh <dig> domain, we find that information between functional sites is conveyed by a contiguous pathway of residues that cross the hydrophobic core to connect both sites.
in particular our results suggest a clear information flux between both binding sites and residues near the linkers of the domain towards the other domains of the fyn kinase.
as a result it is likely that removal of the phosphopeptide from the sh <dig> binding site uncouples the sh <dig> domain from both the sh <dig> domain as well as from the kinase domain, thereby facilitating the activation of the fyn kinase as already suggested in the literature  <cit> .
importantly, communication between functional sites does not necessarily have to occur via a pathway mechanism.
other modes of signal transmission may equally be applicable in which the uncoupling of certain residues upon ligand binding plays a more important role .
for instance, allosteric changes could result in a shift of couplings which could be identifiable by clusters of increased as well as decreased mutual information .
using experimental methods such as nmr in combination with protein engineering approaches together with our method for information quantification should allow to test and explicit signal transduction mechanisms for a variety of model systems, which would be highly valuable especially for modular domains such as sh <dig>  sh <dig> or pdz domains.
especially for the latter domain a number of experimental studies have already shown the presence of long-range communication  <cit>  finally, it could contribute to understand how disease mutants and snps modulate the efficiency of information transfer by signaling proteins and whether for certain mutations this correlates with disease penetrance

 cooperative protein response and thus cooperative network behavior requires information transfer between distal sites in a protein or protein complex. protein structures often achieve such long range communication by allosteric movement, but this is certainly not a requirement. essentially any change in the dynamic properties of protein residues, upon ligand binding for example, that efficiently propagates through the structure and is detectable at a distal site, constitutes a form of signal transduction. many theoretical and experimental studies have been devoted to the structural dynamics of globular proteins and its implications for protein function. recent experimental studies, on the one hand, focus on the role of protein dynamics for catalysis, signal transduction, cooperative response and protein aggregation. on the other hand, theoretical approaches have probed the relationship between protein structural dynamics and signal transduction, limited only by the combinatorial complexity associated with the conformational space. in response, techniques such as targeted molecular dynamics, anisotropic thermal diffusion or go-like sampling were developed to reduce the degrees of freedom so as to allow the mapping of a pathway that connects one ground state to its allosteric counterpart. additionally, ranganathan and co-workers have mapped residues that participate in signal transduction in several important proteins by extracting evolutionary correlated mutations from multiple sequence alignments. together these studies show that changes in protein dynamics can propagate through the protein structure thereby creating long range correlations between distal active sites, only a fraction of residues in a protein structure participate in signal propagation, and these intra-protein communication modes are generally conserved within protein families and even protein folds. now that the principles relating correlated structural dynamics to signal transduction mechanisms within proteins are becoming apparent, we here present a method to identify and quantify these structural fluctuations in terms of information allowing computation of the information transfer between active sites. in particular, we use information theory, and more specifically the concept of mutual information, as a means to describe the relationship between structural protein dynamics and signaling behavior. this association directly follows from the relation between the definitions of entropy as a measure of structural disorder in statistical thermodynamics, on the one hand, and as a measure of error on communication channels in information theory on the other hand. the potential of globular proteins to convey information throughout their structure, thereby correlating the behavior of distant effector sites, is indeed provided by the change in structural dynamics induced by ligand binding. but more fundamentally, information transfer originates from mutual conformational dependence of the different residues composing a protein or protein complex. in other words, when the conformational flexibility of one residue affects the conformational flexibility of another residue, changes in the structural dynamics will produce correlated changes in entropy and information is thus exchanged. mutual information quantifies the amount of conformational dependence between protein residues. as a result a protein can be considered as a network of residues exchanging conformational information. this interpretation provides a theoretical framework to understand how the redistribution of thermodynamic perturbations within a network of protein residues constitutes an information transfer network. to illustrate our information theoretic approach, we map the residue-based information transfer network of the sh domain of the fyn tyrosine kinase, a member of the src family of kinases, that are involved in essential signaling pathways, for example controlling cell growth, proliferation, differentiation, motility and cell adhesion. family members share a common multi-domain architecture, with the archetypal src homology domain at the n-terminal, a src homology domain, the tyrosine kinase domain and a short c-terminal tail. in its inactive form the kinase domain is sequestered by the sh and sh domains. the sh domain docks to the linker connecting the sh to the kinase domain. the sh domain holds on to a c-terminal phosphotyrosine sequence. dephosphorylation of this c-terminal sequence leads to its release from the sh phosphopeptide-binding site. this in turn leads to the release of the sh2-kinase linker from the sh domain and undocking of the kinase domain which is more than angstrom away from the sh phosphopeptide binding site. the sh domain has a typical length of approximately amino acids and displays a simple fold in which a central anti-parallel β-sheet is flanked by two alpha-helices. the β-strands are commonly annotated from βa to βg, whereas the α-helices are referred to as αa and αb. panel b of figure shows the labeled structure of the fyn sh domain that is used in the present analysis. the peptide binding site of the fyn sh domain consists of two functional regions: the phosphotyrosine binding pocket, responsible for ptyr recognition, and the hydrophobic binding pocket, which interacts with the residues c-terminal to the ptyr. the first binding cavity is located between the central anti-parallel β-sheet and the α-helix αa. the second binding pocket is located on the opposite side of the phosphotyrosine binding cavity, between the central β-sheet and the α-helix αb. in figure 2c, the names of the residues involved in binding the c-terminal phosphorylated sequences are annotated onto the structure following the nomenclature defined in. these names, e.g. hisβd will be used when referring to residues of either binding cavity. other residues, e.g. ser will be indicated by their amino acid name and position in the fyn sh pdb structures used here. for further detailed discussion of sh function and nomenclature we refer to the literature. experimental work on the role of the linker between the sh and sh domain of fyn has shown that the nature of the linker, i.e. the residue pattern, determines both orientation and coupling between the two domains. as such it has a direct effect on the activation behavior of the protein kinase and on the intra-molecular communication. this observation is supported by experimental and molecular dynamics work performed by kuriyan et al. on the src kinase. they showed that the release of the c-terminal sequence from the sh binding site uncouples the motions of the sh and sh domains by increasing the flexibility of the linker loop connecting these two domains. the connector link between the sh and sh domain therefore acts as an inducible snap lock mechanism and replacement of three of the eight residues in this linker by glycines generates a constitutively active kinase. in effect, the phosphopeptide binding site at one end of the sh domain is thus coupled to the sh3-sh linker at the opposite side of the sh domain. importantly, cooperativity is achieved without major allosteric movement within the sh domain and therefore originates from subtle differences in dynamics between phosphopeptide bound and unbound states. our analysis shows that mutual information identifies and quantifies the cooperativity in the sh domain over long distances, confirming other results for different non-allosteric structures. in other words, binding the phosphotyrosine peptide triggers an information exchange from the sh binding pockets toward residues located at the opposite side of the domain, targeting the linker regions. hence, the sh peptide binding site acts as a switch that distributes information on its binding state to both linkers with the other domains, coordinating the sh2-sh docking module.  how to quantify the information exchange within a protein? in shannon's theory of information transduction along noisy channels, where the level of noise corresponds to the fraction of the input characters that may be wrongly translated to output characters, information is quantified as the degree of certainty obtainable about the output signal, once a particular input is given. a random channel that produces for any input signal a random output has a maximum degree of uncertainty, whereas for a noiseless channel, which produces an exact copy of the input at the output, there is no uncertainty about the output. this variation in the degree of uncertainty is expressed in terms of entropy, allowing for the immediate application of information theoretical principles to protein dynamics simulations in a straightforward manner. specifically, we focus here on the notion of mutual information to express the mutual dependence between the conformational states of residues in a protein, as a quantification of the information flow within. for more information on information theory we refer to. as a tool, mutual information, and other co-variation measures, has been applied to identify co-varying or co-evolving residues in multi-sequence alignments. yet, as far as we are aware, no work exists that applied this informational theoretical measure to determine the conformation coupling between residue sidechains. so when do two sidechains exchange information? two linked residues are dependent when knowing one conformation will convey information about the conformation of the other. in this case, the two residues share information. for instance, consider two amino acid residues in a protein, each having some conformational variation in their side chains. the mutual information shared by these residues is defined by the entropy reduction that is observed at the second residue when the conformation of the first is fixed in an arbitrary configuration. thus, when ligand binding conformationally restricts the first residue then mutual information will quantify how much of the signal of ligand binding is received at the second residue since its conformational flexibility will also be restricted. mutual information between two residues is thus zero when the conformational state of one residue does not provide any information about the conformational state of the other residue. on the other hand, mutual information is maximal when each conformational state of a residue's sidechain uniquely defines the conformational state of the other residue's sidechain. in accordance with information theory, the conformational space of each residue corresponds to the residue's alphabet. exploration of the conformational space of the amino acids the calculation of mutual information between any two amino acids in a protein requires reliable statistics of the conformational dynamics of all amino acids in the native structure of the protein. the method relies on computing both the probability of residue i to adopt an arbitrary conformation independently and the joint probability of finding residue i and residue j in an arbitray combined combination. this is achieved via effectively sampling the conformational states of each amino acid in a protein structure so that all possible combinations of conformations of residues i and j can be explored. it is clear that the computational size of such a sampling problem can only be addressed by reducing the number of possible conformations that each amino acid can adopt to a relatively small number of discrete states. to achieve this, we treat the sidechain and backbone flexibility of each amino acid as separate components of its overall conformational dynamics and consider only a finite number of states for each. in order to keep the number of sidechain conformations computationally tractable, it is common to employ a discrete, finite-size alphabet for each residue, called rotamer library. such rotamer libraries are constructed by extracting from a database of high resolution protein structures the most frequently occurring states and thus the degree of coarse graining that is employed to record the statistics imposes a resolution on the data. for the purpose of detecting conformational dependencies between neighboring residues in a protein structure, we require a finer resolution than is provided by common rotamer libraries used for homology modeling and we thus constructed a database of backbone dependent sidechain conformations with a degree resolution on the sidechain dihedral angles. given the backbone dihedral angles of a certain residue in the protein structure, a list of possible sidechain conformations and their probabilities can be retrieved from this database. the number of chi angles in the residue determines the size of this list, meaning that residues with small sidechains tend to have shorter lists than residues with long sidechains. the conformational state of the sidechain in the protein structure serves as a starting point for generating these lists. information concerning the flexibility of the backbone of a protein can be obtained from either molecular dynamics simulations or nmr data. exploration of the conformational space accessible to the protein backbone using md is a computationally expensive method. moreover, vendruscolo and co-workers have recently shown that md simulations yield more realistic results when restricted by experimental data in the form of residue-residue distances derived from nuclear overhauser effects in a nuclear magnetic resonance experiment, suggesting that unrestricted md simulations are not the most effective method to sample the backbone ensemble. the work of vendrusculo and others also revealed that the backbone dynamics can essentially be captured by a small number of representative backbone structures. moreover, bahar and colleagues recently showed that nmr models can also be viewed as an ensemble of conformations accessible under physiological conditions and that even though the rmsd values reflect the uncertainties in the coordinates, they also contain physically meaningful contributions of equilibrium fluctuations. although this point is still under debate, we here found that sampling the sidechain conformations of each amino acid on the collection of backbones present in the nmr dataset using a monte carlo approach yields results that are consistent with experimental studies as well as more exhaustive simulations performed by kuriyan et al. more specifically, we employed a metropolis algorithm implemented in the foldx force field from which an equilibrium distribution of sidechain conformations, compatible with a given protein backbone structure, is obtained. the force field scores the conformations taking into account the packing interactions. the sidechain sampling results for a single backbone structure obviously introduce a strong bias in the apparent information flux towards residues whose sidechains arbitrarily happen to be strongly coupled in this particular backbone conformation. when information is obtained from an ensemble of related backbone conformations however, each backbone introduces slight variations in the pattern of residue-residue couplings. as a result, the combination of the sidechain sampling on the entire ensemble of backbone structures acts as a filter that removes sporadic couplings while accumulating consistent couplings, thereby revealing the true network of information exchange between all residues. we have here taken the entire ensemble of backbone structures in the nmr datasets of the fyn kinase sh domain as an adequate sample and have not systematically explored if a reduced number of backbones could be employed to the same effect. construction of the residue-residue information network in fyn sh2 since the monte carlo sampling directly yields the equilibrium distribution of sidechain conformations observed at each position along the protein backbone, the quantity of mutual information between all residue pairs in the protein can be calculated using probability and entropy calculations. in this way, information transfer over both short and long distances in the folded protein is elucidated, explicitly revealing the communication between all residues in the protein. extracting only the changes in mutual information that result from the difference between bound and unbound states of the sh domain, highlights the change in communication patterns. hence, those amino acids that experience the strongest changes in mutual information can be detected and mapped on the network of sidechain interactions in the protein, thereby revealing how series of dynamically coupled residues in the topology of the protein direct the overall communication. in order to obtain the network of residue-residue couplings in the fyn sh domain, we employed a foldx based sidechain sampling on the backbone structures of this sh domain as determined by nmr on the protein domain in isolation and bound to its phosphopeptide ligand. the nmr ensemble was determined using over 90% of the structurally non-redundant nuclear overhauser effects, so that the ensemble can be considered to possess adequate precision and accuracy. in order to ensure reliable statistics, we collected over thousand samples of sidechain configurations from approximately million simulation steps over the backbone models present in the nmr ensemble, producing the probabilities of finding the residues' sidechain in a particular conformation. from all these probabilities mutual information between all residue pairs was derived. to capture how the information exchange within the sh domain changes due to ligand binding we require nmr structural data on both the bound and unbound state of the protein. since there is no structural data on the unbound state of fyn sh available publically, we assume here that the backbone flexibility for bound and unbound state can be derived from the ensemble stored in the 1aou dataset by energy minimization. this simplification has as result that large conformational changes in the backbone of the sh domain are not taken into account. yet since domains do normally not experience large structural changes, this simplification may still provide viable results. changes in mutual information link residues at long distances to visualise the mechanistic implications for our model system a clustering algorithm was applied on the mutual information matrix in figure 2a and mapped on the structure of fyn sh figure 2c displays the residue cluster for which the mutual information increases most upon peptide binding. only those pairs of residue whose mutual dependence is above this threshold are accepted in the cluster, meaning that they form a complete graph where the weight of each link is above the threshold value. on the one hand, and as expected, these are composed of residues directly interacting with the phosphopeptide itself but interestingly also consists of residues on the other side of the domain from which the sh2-sh and sh2-kinase linkers emerge. even though previous studies of the fyn sh domain did not assign an important role to the conserved hisβd our analysis suggests this residue experiences a very strong change in mutual information upon ligand binding with all other residues in the structure. this strong effect may be the result of the displacement of the helix αa in this particular sh domain, making the ptyr-binding cavity narrower when the peptide is bound. most affected seems to be residue thref thus, our analysis reveals a strong coupling between the peptide binding site of fyn sh and its sh3-sh and sh2-kinase linkers. these first results show that the sampling approach discussed here provides a direct way to identify and quantify highly coupled groups of residues. moreover, it confirms the notion that subtle changes in structural dynamics can effectively couple residues at distal locations in the structure. note that the clustering does not show how this information exchange actually occurs, i.e. it does not provide a causal explanation. it provides an identification of the residues that may be involved with signaling. yet, the idea is that when, by lowering the threshold, a coherent collection is obtained, possibly identifying a consecutive path that links the binding region to other parts of the domain structure. closer inspection of the information exchange for some residues to understand the key elements of information exchange in proteins we compared the change in mutual information of both silent and informative residues with the rest of the protein. we will here illustrate our analysis by two examples that recapitulate the key features of both informative as well as silent residues. ser will serve as an instance of the collection of silent residues whereas hisβd represents a prominent member of the informative residues. specifically, we mapped the change in mutual information of these residues with all the other residues of the sh domain onto the structure. strong changes are colored red, weak changes blue and intermediate changes are white. in figure 3a, we observe for hisβd that three classes of residues contribute significantly to its score. first, locally interacting residues directly influence each other. second, the phosphorylated peptide connects all the relevant residues in the binding pocket, creating a channel by which conformational fluctuations can be transmitted. as a consequence, these residues are mutually dependent on the state of hisβd third, hisβd is also coupled to a number of residues at the other side of the protein. we see in figure 3a that the residues at the kinase linker that were previously detected by clustering are again present, i.e. long-range communication between the histidine and residues val pro and val is clearly visible here. moreover we see also a coupling with a number of residues related to the sh3-sh2-linker region, meaning that information is exchanged with both linker regions when switching between unbound and bound state. the results for ser are strikingly different. in figure 3b, we see that when the phosphorylated peptide becomes attached to the sh domain, only a few residues located in the peptide binding site exchange more information with this particular residue. further, informative and silent residues do not only strongly differ in terms of the amount of residues with whom they are coupled but also in terms of the distances over which they communicate. figure shows that whereas ser is conformationally isolated, hisβd has an extensive network of couplings with both proximate as well as distal residues. contiguous communication pathway in fyn sh2 as argued in previous sections, our information theoretical approach provides a method to quantify the change in exchange of mutual information between all pairs of residues of the fyn sh domain as a consequence of ligand binding. our analysis revealed that for the majority of residues the conformational coupling to the rest of the protein is not altered upon peptide binding and as a consequence they can be considered silent in terms of signal transduction. a small fraction of residues, however, experience a significant change in conformational coupling and are thus information-rich. clustering only the most informative of these residues revealed a strong coupling between the peptide binding site and the region harboring the linkers connecting the fyn sh to the other domains of the fyn kinase. it remains of course to be explained by which structural mechanism these distal sites become coupled. as argued earlier, the current framework can only identify and quantify the residues that are involved in the interaction. to identify the causal relationships a different analysis is required. several mechanisms for signal transduction have been described in the literature and as such our method does not provide the means to unequivocally single out a given mechanism. however, clustering the informative residues and mapping these on the structure of the protein can provide substantial information. for instance the relative occurrence of coupling versus uncoupling pairs and their disposition on the topology of a protein domain could allow to distinguish pathway models from allosteric models. as our model only displays increase in conformational coupling upon ligand binding, suggesting a pathway model, we here extract the group of residues that define the pathway through clustering of informative residues that have a mutual affinity higher than a particular noise level. in figure we see how the pathway is formed by decreasing the clustering threshold from to bits: a contiguous dynamic pathway emerges, involving the previously identified residues in the peptide binding site and the linker region. as discussed above only a subset of residues of the sh domain is involved in signal transduction, whereas the other residues seem to be conformationally isolated from ligand binding. importantly, signal transduction is achieved throughout the core of the sh domain and involves the main secondary structure elements of the structure. hence, we show that the fyn sh domain acts as indivisible information transmission unit propagating information from its binding site over the tertiary structures to the linker regions of the domain.  driven by the availability of genomics and proteomics data, biological research is increasingly focused on cellular networks rather than on individual proteins. accordingly, a major effort in systems biology is devoted to understanding how topology shapes macromolecular networks into functional units performing information processing tasks such as logical gating, band-pass filtering or signal modulation. biological networks are generally represented by graphs of interconnected protein nodes. in reality, however, these networks form macromolecular complexes whose assembly is determined by intermolecular atomic interactions. as shown by several experimental and theoretical studies, ligand binding may induce changes in the structural dynamics of proteins. these perturbations propagate cooperatively throughout the protein, coupling distal locations and thereby effectively transferring information. hence, proteins domains are the elementary units of information processing in cellular organisms. as such proteins can themselves be represented as networks of interconnected residues exchanging information. protein complexes can by extension be modeled in a similar way. the method introduced here allows to quantitatively translate protein structural dynamics in terms of shannon's information theory. moreover, mutual information permits to determine how much information is exchanged between each pair of residues within a protein structure. transposing structural dynamics in terms of shannon information has the advantage of providing a description level that matches the relevant functional features – i.e.biological information,- that we aim to extract from signal transduction pathways. moreover, the analysis does not make any assumptions on the information transmission model. it strictly focuses on identifying the residues whose coupling increases or decreases as a consequence of peptide binding. further, the method is applicable in identical manner to residue networks at atomic scale as well as to macroscopic descriptions of cellular networks. as explained in results, the mutual information exchanged between residues in protein residue networks is simply a statistical thermodynamics description of the conformational coupling between these residues. by discretising conformational space at the residue level, we easily find the average shannon information for each residue by calculating the average entropy of the conformational probability distribution for each residue. the mutual information exchanged between two residues is the difference of the shannon entropy from one residue and the conditional shannon entropy of that residue given knowledge of the probability distribution of the other residue. in other words, it represents the amount of coupling between two protein residues. once the mutual information is calculated for each pair of residues, clustering can be applied to determine the structural mechanism by which information is transferred between functional sites. in effect we hereby consider a protein as an instance of a noisy coding channel linking a probability distribution of input signals at one binding site with a probability distribution of output signals at another binding site. for the fyn sh domain, we find that information between functional sites is conveyed by a contiguous pathway of residues that cross the hydrophobic core to connect both sites. in particular our results suggest a clear information flux between both binding sites and residues near the linkers of the domain towards the other domains of the fyn kinase. as a result it is likely that removal of the phosphopeptide from the sh binding site uncouples the sh domain from both the sh domain as well as from the kinase domain, thereby facilitating the activation of the fyn kinase as already suggested in the literature. importantly, communication between functional sites does not necessarily have to occur via a pathway mechanism. other modes of signal transmission may equally be applicable in which the uncoupling of certain residues upon ligand binding plays a more important role. for instance, allosteric changes could result in a shift of couplings which could be identifiable by clusters of increased as well as decreased mutual information. using experimental methods such as nmr in combination with protein engineering approaches together with our method for information quantification should allow to test and explicit signal transduction mechanisms for a variety of model systems, which would be highly valuable especially for modular domains such as sh sh or pdz domains. especially for the latter domain a number of experimental studies have already shown the presence of long-range communication finally, it could contribute to understand how disease mutants and snps modulate the efficiency of information transfer by signaling proteins and whether for certain mutations this correlates with disease penetrance 
mapping out changes of mutual information on the protein structure then allows visualizing how distal communication is achieved. as a result, our method provides a means to directly map the exchange of biological information on the structure of protein domains, making it clear how binding triggers conformational changes in the protein structure. as such it provides a structural road, next to the existing attempts at sequence level, to predict long-range interactions within protein structures. although often associated with large allosteric movements, more subtle changes in protein dynamics can also induce long-range correlations. here we introduce a method to analyze protein dynamics within the framework of information theory and show that signal transduction within proteins can be considered as a particular instance of communication over a noisy channel. our analysis reveals that the fyn sh domain forms a noisy communication channel that couples residues located in the phosphopeptide and specificity binding sites and a number of residues at the other side of the domain near the linkers that connect the sh domain to the sh and kinase domains. we find that for this particular domain, communication is affected by a series of contiguous residues that connect distal sites by crossing the core of the sh domain. in particular, we analyze the conformational correlations between protein residues and apply the concept of mutual information to quantify information exchange. however, an appropriate formalism that directly relates protein structural dynamics to information exchange between functional sites is still lacking.
 efficient communication between distant sites within a protein is essential for cooperative biological response. although often associated with large allosteric movements, more subtle changes in protein dynamics can also induce long-range correlations. however, an appropriate formalism that directly relates protein structural dynamics to information exchange between functional sites is still lacking.  here we introduce a method to analyze protein dynamics within the framework of information theory and show that signal transduction within proteins can be considered as a particular instance of communication over a noisy channel. in particular, we analyze the conformational correlations between protein residues and apply the concept of mutual information to quantify information exchange. mapping out changes of mutual information on the protein structure then allows visualizing how distal communication is achieved. we illustrate the approach by analyzing information transfer by the sh domain of fyn tyrosine kinase, obtained from monte carlo dynamics simulations. our analysis reveals that the fyn sh domain forms a noisy communication channel that couples residues located in the phosphopeptide and specificity binding sites and a number of residues at the other side of the domain near the linkers that connect the sh domain to the sh and kinase domains. we find that for this particular domain, communication is affected by a series of contiguous residues that connect distal sites by crossing the core of the sh domain.  as a result, our method provides a means to directly map the exchange of biological information on the structure of protein domains, making it clear how binding triggers conformational changes in the protein structure. as such it provides a structural road, next to the existing attempts at sequence level, to predict long-range interactions within protein structures. 