The importance of dynamic effects of allosteric drug design are now being recognized. Allosteric communication derives from the propagation of effects between distant points in a protein and is closely related to transfer of entropy between these points. In this communication, two methods of evaluating entropy transfer between residues are presented.The first method isbased on extensive molecular dynamics simulations and the second, a fast and approximate one, based on the coarse-grained Gaussian Network Model of proteins. In this brief communication, we discuss two examples, one each of these two methods and determine the extent of entropic coupling between residues resulting from binding of a substrate. Predictions of both models on allosteric communication patterns show that transfer of information in proteins does not necessarily follow a single path, but involves essentially the full protein. It is also shown that considering entropy only is not sufficient to detect allosteric communication and additional information based on time delayed correlations should be introduced. We show that binding of a substrate to the protein introduces new inter-residue correlations to the protein and changes the entropy transfer features of residues. The decrease in the entropy or the level of uncertainty in the fluctuations of residues upon binding is expected to be of great importance for allosteric drug design.The two models that we discuss are (i) ubiquitin and its complex with a substrate, and (ii) the PDZ domain and its behavior in the presence and absence of its C-terminal tail.

•    Molecular dynamics
•    Dynamic gaussian network model
•    Transfer entropy
•    Drug design
•    ubiquitin
•    PDZ domain

Hacisuleyman A, Erman B (2017) Entropy Transfer and allosteric Communication in Proteins. J Drug Des Res 4(5): 1054.

Drug action on a protein is now known to be strongly coupled with the conformational preferences andtransitions in the protein that result from dynamic fluctuations of the protein and regulate its allosteric characteristics. Proteins are not static objects. Each residue fluctuates around a well-defined mean position, and the fluctuations of residues are correlated with each other, leading to a coherent dynamic object. Furthermore, perturbation of a given residue by an external source, such as a drug molecule for example, results in changes in the amplitudes and rates of fluctuations of other residues which is one possiblesource of allostery in proteins. Allostery results from an observed shift in conformational preference in the protein associated with atomic fluctuations. This leadsto vibrational entropychangeswhere a changein fluctuations on a site is compensated by a correlated change of fluctuations on another distant site. Transmission of effects through correlated fluctuations of residues is a universal property of all proteins and not only of allosteric ones. In this sense, all proteins may be regarded as intrinsically allosteric in nature [1-3]. This expanded view on allostery allows for the consideration of entropy transfer in proteins rather than the former limited picture of discrete two state transitions [4]. Allosteric communication first requires the identification of two sites, the effector site, i.e., the site that is acted upon, and the regulatory site where protein’s activity is regulated. Most known cancers result from disruption or alteration of allosteric communication resulting from single mutations. The solution of the problem can be narrowed down to finding whether a given pair of residuescommunicate with each other, and if so what the consequences of this communication are. Stated in another way, how does the communication patterns of a protein change? Various approaches exist for understanding changes in correlated motions. The idea of transfer entropy, introduced by Schreiber [5], is an appropriate one for understanding communication and information transfer in proteins.The problem is simply stated as follows: Suppose we know the time trajectories of two residues, i and j. These trajectories are subject to the noise-like effects of the environment. And we also assume that there is some correlation between the two trajectories i and j. We assume that the present values of j are predicted by its past values. This knowledge causes a decrease in the uncertainty of the present values of j. Now, if the past values of i also affect the present values of j and decrease the uncertainty of the trajectory j, then we say there is a transfer of information from the past values of i to the present values of j. Schreiber called this transfer entropy. Transfer entropy is a measure of the dynamics of communication in a system. Here, we adopt the term ‘allosteric communication’. The fact that entropy transfer is based on correlations between past and present states of residues, we conclude that allosteric communication cannot be estimated by time independent static correlations. Thus, the knowledge of the amplitudes of correlations between the fluctuations of two residues is not sufficient to describe allosteric communication, and information on decay of correlations is also needed. We will discuss this point further in the remaining part of the paper. The concept of entropy transfer in proteins is relatively new. van der Vaart applied the Schreiber equation to determine information flow between Ets-1 transcription factor and its binding partner DNA [6], [7], quantified entropy transfer among several residues in a molecular dynamics analysis of mutation effects on autophosphorylation of ERK2 [8], analyzed entropy transfer in antibody antigen interactions [9], applied the method to understand changes in correlated motions of the Rho GTPase binding domain during dimerization. In a recent paper [2], we used the Schreiber transfer entropy concept to analyze ubiquitin. In a subsequent paper [3], we developed a new model of entropy transfer based on rapid but approximate calculation of entropy transfer by the dynamic Gaussian Network Model of proteins. The present short communication discusses and clarifies the findings of two systems analyzed in those two references. Here, we discuss the increase in the dynamic coherence of protein structure when a substrate binds to the protein. Coherence is a consequence of entropy transfer between residues and is a measure of the decrease in the uncertainty of the behavior of residues. We would like to point out that an increase in coherence does not necessarily imply an improvement in desirable properties of the protein. For example, if binding of a substrate increases the correlations already present in the unbound state, then our knowledge of the motion of residues will change, but this does not imply that this change improves the system towards a desired state or not. We elaborate on this in discussing the two examples that we already used ubiquitin and the PDZ domain. In this respect, the present short communication may be regarded as a reinterpretation of our two earlier papers.

Currently, one way of calculating transfer entropy in proteins is to run molecular dynamics simulations in the order of microseconds and considerthe time trajectory of the fluctuations of two residues. The two residues may be spatially distant. We search for information transfer from the trajectory of one residue to that of the other. For information to be transferred from an atom i to another atom j, their trajectories should be correlated and this transfer should be asymmetric, i.e., information from i to j should not be equal to information going from j to i. This requirement brings up the use of time delayed correlations of fluctuations which are asymmetric and accounts for directionality of information flow and therefore causality in contrast to time independent correlations. If Cij(t,t+τ) denotes the correlation of fluctuations of i at time t with those of j at time t+τ, then asymmetry requires that Cij(t,t+τ)≠ Cji(t,t+τ). This introduces directionality and therefore causality. If the time delayed correlations are asymmetric, net information transferred from an atom ito another distant atom j can be quantified if the problem is posed in terms of entropy transfer. Through analysis of entropy transfer, residues that act as drivers of the fluctuations of other residues, the entropy sources of the protein, can be determined; thereby causality inherent within the correlations can be detected. Determining residues that act as drivers and those that are driven is important especially from the point of view of drug design.

General equation of entropy transfer: In order to facilitate the discussion, we first discuss transfer entropy in terms of molecular dynamics trajectories. We perform molecular dynamics simulations for a protein in equilibrium and extract trajectories for each atom. The fluctuation, ΔRi of the position vector Ri of a residue from its mean position Ri is a rapidly changing function of time. We focus only on the alpha carbon of each residue. Transfer entropy ( ) Ti j → τ from the trajectory ΔRi of residue i to the trajectory ΔRj of residue j is the amount of uncertainty reduced in future values of ΔRj (t+τ) by knowing the past values of ΔRi (t) for the given past values of ΔRj (t). The amount of information is measured using Shannon’s entropy and according to Schreiber’s work, transfer entropy from atom i to atom j with a time delay of τ can be written as

Here, S(ΔRj (t+τ)|ΔRj (t)) is the conditional entropy of having a fluctuation ΔRj (t+τ) at time t+τ, given that the fluctuation is ΔRj (t) at time t.S(ΔRj (t+τ)|ΔRi (t)ΔRj (t)) is the conditional entropy of having a fluctuation ΔRj (t+τ) at time t+τ, given the fluctuations ΔRj (t) and ΔRi (t)at time t. The latter term shows the effect of residue i on residue j. If residues i and j are uncorrelated the entropy transfer term, Eq. 1 will be zero. If correlated, then the earlier knowledge of ΔRi (t) will reduce the entropy S(ΔRj (t+τ)|ΔRi (t) ΔRj (t))below S(ΔRj (t+τ)|ΔRj (t))which means that the uncertainty in the trajectory of j will be reduced. Thus ( ) Ti j → τ is a measure of the reduction of uncertainty in the fluctuations if residue j.

When written in terms of probabilities

(Note: A typo in Eq, 15 of the original paper [3] is corrected here) Conditional entropy of atom j at t+τ given j at time t will provide the effect of the prior fluctuations of the j th atomon its future values, and the triple probabilityterm will provide the effect of prior fluctuations of atom i and j on future fluctuations of atom j. When combined, these two terms will help us quantify solely the effect of i th atom’s fluctuations on future values of atom j. In the present work, trajectories of atomic coordinates were generated by using 1 microsecond MD simulations from which probabilities of atomic coordinate fluctuations are calculated. The magnitudes of fluctuationsarepartitioned into discrete bins. Each bin is referred to as a state. The probability of the fluctuation to be in state i is obtained by dividing the number of occurrences in the ith state by the total number of occurrences and is denoted by pi . Details of calculations of probabilities may be found in our previous work [3].

Entropy transfer helps us to detect the information gained when we know the correlations between the residues. Time delayed correlations help us to determine which residue drives this correlationand which is being driven. If the fluctuations of residue i control the fluctuations of residue j, then we say residue j is driven by residue i, the decay time for Cij(τ) will be larger than the decay time for Cji(τ) [10].

The Dynamic Gaussian Network Model: Using molecular dynamics to generate trajectories becomes a serious bottleneck and a rapid characterization is required. To provide a rapid scheme of computing entropy transfer in proteins we formulate [2], transfer entropy using the dynamic version of the Gaussian Network Model (dGNM) which considers harmonic interactions between contacting pairs of residues. In this scheme, the only input to the system is the three-dimensional structure of the protein using which we count the number of neighbors of each residue and form the connectivity matrix (See reference [11]. No trajectories are required for the calculation of probabilities (See below) and calculation times for determining transfer entropy between all pairs of residues of proteins, even of extremely large protein complexes can now be performed in the order of seconds on a laptop computer using dGNM.

The spring constant matrix, ?, where a spring of constant unity is assumed between residues in contact forms the basis of the dGNM. It is defined as follows: ?ij equates to −1 if alpha carbons of residues i and j are within a cutoff distance of rc and to zero otherwise. Each i th diagonal element ?ij is equal to the negative sum of the i th row. The time dependent correlation of fluctuations is given as (Haliloglu, Erman, & Bahar, 1992)

Where λk is the kth eigenvalue and ui (k)is the ith component of the kth eigenvector of the ? matrix. Full details of the solution are explained in [12]. Knowing the correlations shown in Eq. 3, the probabilities given in Eq. 2 may now be evaluated [2] and entropy transfer can ve written as

Equations 2 and 4 give us the amount of uncertainty reduced in the motions of a residue j at a given time, resulting from its coupling with residue i at an earlier time. The coupling of residue jto all residues of the protein, which we denote by Tj is obtained by summing Equations 2 and 4 over i, i.e.,

By definition, ( ) Ti j → τ is positive. Therefore, Tj is positive,

i.e., correlations always decreasing the transfer of entropy from the protein to the given residue? The aim of this paper is to show how binding of a substrate changes the communication patterns of the protein.

In the two examples that we give below, we focus on Eq. 5 which gives the important information on the degree of coupling of a residue to the rest of the protein, and we discuss the relevance of this to allosteric communication.

Role of correlations on information transfer: Due to the myriad of effects acting on a given residue, there will be an uncertainty associated with the knowledge of its position and its instantaneous fluctuation. There will be a similar uncertainty in the fluctuations of the second residue.If the two fluctuations are not correlated, then, knowing the fluctuations of the first residue will not reduce the uncertainty in the trajectory of the second residue. The other extreme case is where both fluctuations are perfectly locked into each other, then the knowledge of the first trajectory will leave no uncertainty in the second trajectory and we will have full information on the second trajectory. In the presence of correlations, there will be an information transfer from the first residue to the second. If an observation is made at time t=t on the first trajectory and another observation attime t=t+τ on the second trajectory, then the amount ofinformation gained on the second trajectory by the knowledge of the first willdecay depending on thelength of τ and the level of correlation between the two trajectories. The time dependent correlation of two fluctuations is expressed in terms of the time delayed correlation function , which shows the scalar product of the two vectors and the angular brackets denote an average over all observations, with a time delay of τ between the two. If this average is not zero, then we see that the fluctuations of i at one time correlate with the fluctuations of j at a time τ later. If the two residues are symmetrically correlated, that is, = , then there will be no net informationtransfer from one residue to the other, because effects propagate symmetrically between the two residues. Net information may be transferred from one residue to the other only if ≠ . It is to be noted that time independent correlations, , are always symmetric in i and j. Asymmetry is possible only in time delayed correlations.

Example 1: Effect of substrate binding on entropy transfer in Ubiquitin: Ubiquitin is a 76 residue protein, shown in the left panel of Figure 1, which is known to propagate signals allosterically in the cell by binding to a vast number of substrates [13].

Figure 1: The structure of a) Ubiquitin (1UBQ) is shown in the left panel and b) Ubiquitin-Protease complex (2KTF) is shown in the right panel. The switch residues GLU24 and GLY53 are shown as ball structures on the left panel.

Ubiquitinproteasome system is involved in many cellular processes such as degradation. In a recent study by [14] a collective global motion is identified which originates from a conformational switch formed by GLU24 and GLY53.The switch residues are located far from the binding site residues of Ubiquitin, as shown on the left panel of Figure 1. Here we show that binding of the protease, 2L0G, causes a significant reduction in the entropy of the binding site residues ILE36, GLN49, LEU71-ARG74 of Ubiquitin, as well as in a significant decrease of entropy of the switch residues GLU24 and GLY53. According to [14], the allosteric communication of Ubiquitin is provided by the dynamics of GLU24 and GLY53. The strong decrease in the entropy of these two residues indicates that binding results in important changes in the allosteric behavior.

Figure 2: Reduction in entropy of residues (shown along the abscissa) due to their couplings with the remaining residues. Light curve, obtained from Eq. 5, shows only Ubiquitin and the heavy curve shows the entropy reduction in Ubiquitin-Protease complex.

In Figure 2 we show the reduction in entropy transfer of residues before and after forming a complex with the protease, 2KTF. The empty arrows locate GLU24 and GLY53. The filled arrows show the residues of Ubiquitin in contact with the protease. Increased values of Tj due to complex formation indicates that uncertainty is reduced due to coupling of these residues with the rest of the protein. Reduction in uncertainty of ILE3, GLU16, GLU24 and GLY53 are higher in the Ubiquitinprotease complex. ILE3 and GLU16 are on a beta strand, GLU24 is a part of a helix and GLY53 is in a loop making a hydrogen bond with GLU24 when in complex with the protease. ILE3 is on a surface and is susceptible to the binding of a third protein. GLY53 couples with GLU24 and GLU16. The changes in entropy transfer features in Ubiquitin affect the residues of the protein collectively and cannot be seen as effects propagating through a single predetermined path. Ligand binding affects the entropy transfer characteristics of the full protein.

Example 2: Entropy transfer in the PDZ domain. Using the dGNM versions of transfer entropy, we studied entropy transfer in the PDZ domain.PDZ domain proteins act as information transmitters from one protein to another and take place in key cellular processesin the cell and they are known as allosteric proteins. [15] showed evolutionarily conserved networks of allosteric communication in PDZ structures. [16] showed that the removal of a distal alpha helix, α3, did not affect the stability of the protein but resulted in a 21-fold decrease in the binding affinity of the protein to a peptide in a member of the PDZ family, the protein PSD-95. The ribbon structures of PDZ in the presence and absence of the distal helix are shown on the left and the right panels of Figure 3, respectively.

Figure 3: The structure of PSD-95(1BFE) is shown, the left panel is the complete protein and the right panel is the protein without the distal alpha helix, α3.

Results of the removal of this helix show that this helix possesses allosteric activities of entropic origin. Here, we show that the distal alpha helix interacts with the rest of the protein entropically. We also show that removal of this interaction results in severe changes in entropy transfer in the protein Figure 3.

Figure 4: Reduction in entropy of residues (shown along the abscissa) due to their couplings with the remaining residues. Upper curve is obtained from Eq. 5 in the presence of α3 and the lower curve in its absence.

In Figure 4, we show the changes in entropy transferresulting from the presence of the distal helix. The ordinate value corresponding to a given residue j shows the transfer entropy Tj between the protein and that residue. Inthe calculations, τ is taken as the time for correlations to decay to 1/e of their original values. Any perturbation applied to the protein will change the correlations in the protein, and therefore will change the uncertainty in the trajectories. The value of Tj is a good measure of the decrease of uncertainty in the trajectory of residue j due to its coupling to the rest of the protein. The presence of the tail introduces changes in the correlations of the protein. The α3 residues are not shown in the figure. The upper curve is obtained in the presence of α3 and the lower curve is in the absence of it. The presence of α3 introduces significant reduction in the entropy of the residues. ALA 347 makes hydrogen bonds with LEU 323, LEU 349 and SER 350 when α3 helix is present but these hydrogen bonds disappear upon removal of the α3 helix The changes in the entropy reduction of LEU323, LEU 349, GLY 364 and ALA 390 in the presence and absence of α3can are observed from Figure 4. TYR 397, located in the α3 helix provides the communication between this helix and the rest of the protein by packing itself against the core of the protein, and deletion of this helix interrupts this communication Figure 4.

The entropy transfer model summarized in this paper most importantly identifies the residues that are involved in entropy transduction upon a perturbation on the protein and it gives the reduction in the uncertainty of the trajectory of residues resulting from their coupling (correlation) with the rest of the protein. Recent work [17-21] show that allosteric communication is mostly entropic and changes the dynamics of the system. It is plausible to assume that the entropy transfer from i to j affects the local dynamics at j. One example is that binding of fructose 2,6-bisphosphate to the regulatory site of pyruvate kinase cools the enzyme and reduces dynamic movement, particularly of the B-domain [22]. The reduced dynamic movement of the ligand bound form traps the pyruvate kinase in its enzymaticallyactive state with the B-domain acting as a lid to cover the active site. Another example, we discussed here is the binding of the α3 segment to the PDZ domain which decreases the entropy of the protein and therefore affects the motions of the peptide binding domain changing the peptide binding affinity of the protein by 21-fold.

The transfer entropy model that we developed for detecting allosteric communication patterns in proteins measures the amount of information transfer between the trajectories of two atoms; it is built on the transfer entropy concept by Schreiber. This model evaluates the uncertainty reduced in future fluctuations of atom j, by knowing the fluctuations of atoms i and j at the present time. Entropy transfer in proteins involves several residues and does not necessarily take place along a single path. Figure 4 shows that the allosteric effect of α3 results in an overall decrease in entropy which cannot be described by a single path event. According to the model, knowledge of entropy is not sufficient for detecting allosteric communication and additional information on time delayed correlations must be introduced. This leads to the presence of causality in proteins and causality in proteins allows for identifying driver-driven relations for residue pairs. Knowledge of driver-driven relation between residues allows us to determine the residues that should be manipulated to control protein activity. This should be of great significance in the sense of allosteric drug design and for understanding effects of mutations on protein function.

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