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Proc Natl Acad Sci U S A. Dec 11, 2007; 104(50): 19813–19818.
Published online Dec 5, 2007. doi:  10.1073/pnas.0708120104
PMCID: PMC2148381

Principles underlying energetic coupling along an allosteric communication trajectory of a voltage-activated K+ channel


The information flow between distal elements of a protein may rely on allosteric communication trajectories lying along the protein's tertiary or quaternary structure. To unravel the underlying features of energy parsing along allosteric pathways in voltage-gated K+ channels, high-order thermodynamic coupling analysis was performed. We report that such allosteric trajectories are functionally conserved and delineated by well defined boundaries. Moreover, allosteric trajectories assume a hierarchical organization whereby increasingly stronger layers of cooperative residue interactions act to ensure efficient and cooperative long-range coupling between distal channel regions. Such long-range communication is brought about by a coupling of local and global conformational changes, suggesting that the allosteric trajectory also corresponds to a pathway of physical deformation. Supported by theoretical analyses and analogy to studies analyzing the contribution of long-range residue coupling to protein stability, we propose that such experimentally derived trajectory features are a general property of allosterically regulated proteins.

Keywords: conformational changes, cooperativity, double-mutant cycles, Hill coefficient, nonadditivity

Long-range communication between distant functional elements is a fundamental property of many allosteric proteins. Information transfer between such elements may be achieved by propagation of conformational changes through a protein structure, induced by changes in chemical or electrical potential. However, while the structures of stable conformational states of several allosteric proteins are known, the mechanism(s) by which ligand-induced structural changes propagate through the molecule remains elusive. In the absence of extensive experimental data accurately describing such allosteric networks, computational analyses have attempted to provide mechanistic explanations for long-range coupling in proteins. Structure-based computer simulations suggest that conformational changes may propagate signal transmission by a redistribution of native-state conformational ensembles (13). Others suggest that conformational changes may propagate by simple mechanical deformation of the protein structure along pathways of energetic connectivity, comprising adjacent amino acid positions in the tertiary structure (4, 5). Conformational changes are central to the function of voltage-activated potassium channels (Kv), pore-forming proteins that open and close in response to changes in membrane potential. Such conformational transitions regulate the flow of potassium ions across the membrane (69), a process underlying many fundamental biological processes, in particular the generation of nerve and muscle action potentials (10). These conformational changes, moreover, play a fundamental role in mediating coupling between the voltage-sensor, activation gate and selectivity filter elements of Kv channels (69, 1114).

Amenable to rapid and highly accurate functional characterization without the need for protein purification, the Kv channel represents an excellent model for studying the features underlying allosteric communication networks in proteins. Recent systematic energy perturbation analysis of the pore domain of the archetypal Shaker Kv channel (15) revealed that gating-sensitive positions—i.e., those positions where mutation dramatically affects the closed-open equilibrium of the channel—cluster around the channel activation gate and near the selectivity filter functional elements, when mapped onto the closed channel pore structure (Fig. 1A). Other gating-sensitive positions were found to lie along a pathway connecting these two elements (Fig. 1A). Through double-mutant cycle analysis (Fig. 1B) (1618), it was shown that residues along this pathway are energetically coupled, implying that the allosteric pathway connecting the activation gate and selectivity filter can be described as a trajectory of energetic connectivity, in accord with the mechanical view of signal propagation presented by Ranganathan and colleagues (4, 5, 19). Now, we apply high-order thermodynamic coupling analysis of voltage-dependent gating of the Kv channel, serving as a model system, to unravel the mechanism underlying experimentally determined features of energy transduction along allosteric communication trajectories.

Fig. 1.
The allosteric communication network found in the pore domain of the Shaker Kv channel. (A) Gating-sensitive positions of the Shaker Kv channel pore (15), including the PVP activation gate residues (lower horizontal line), form a connected pattern when ...


Allosteric Trajectory Residues Are Functionally Essential.

In analyzing thermodynamic coupling along allosteric communication networks, we treat the gating process of the Kv channel as a two-state thermodynamic process, comprising only closed and open channel states. This approach for channel gating is analogous to the classic Hill approach for analyzing cooperative conformational transitions in ligand-binding allosteric systems. Although the accepted gating model of the Shaker Kv channel is considerably more complicated, involving 15 closed channel states and a single open channel conformation (7), detailed scanning mutagenesis analysis of the pore domain revealed that gating-sensitive pore mutations, including all of those analyzed in the present study, affect only the late concerted pore-opening transition, leading from the last closed state to the open channel state (15). These mutations, therefore, do not change the gating pathway of the channel. Moreover, we have previously demonstrated that the channel-opening free energies derived assuming either a two-state or the accepted 16-state gating models are linearly correlated (15). Consequently, using the two-state model to describe Kv channel gating represents a logical and informative reductionist approach to this molecular event.

What is unique about the relation between residues lining the allosteric trajectory of the Kv channel pore domain? To answer this question, in the context of the thermodynamic channel gating process being considered, it is first necessary to describe ΔGiopen, the contribution of a single residue i to the closed to open equilibrium of the channel. Eq. 1 describes all possible free energy contributions to channel opening associated with residue i and considers the context of all possible structural interactions of this residue with other channel residues (20).

equation image

In the simple case, where residue i is not coupled to any other residue, ΔGiopen = ΔGiopen,intrinsic. If, however, residue i is coupled to other residues, then coupling terms up to the order n, where n corresponds to the total number of protein residues, must be considered. Horovitz and Fersht (17) offer the means to understanding the meaning of these high-order coupling terms. Δ2G(i,j) represents the pairwise (second-order) coupling free energy of any two channel residues i and j. Similarly, Δ3G(i,j)k describes the third-order coupling, reflecting the energetic effect of residue k on the coupling free energy between the residue pair (i, j), whereas Δ4G(i,j),(k,l) defines the magnitude of (fourth-order) coupling between two pairs of interactions, (i, j) and (k, l). Although Eq. 1 should ideally extend to ΔnG—i.e., the nth-order coupling of n channel residues—it is reasonable to assume that for most protein residues, the enormous complexity of energy parsing for any thermodynamic process can usually be reduced to only the first few coupling terms, involving only those residues immediately adjacent to the residue in question (21). By contrast, residues involved in long-range coupling would be expected to make significant contributions to higher-order coupling terms.

To address interactions between residues lining the allosteric trajectory of the Shaker K+ channel, including A391, E395, T469, A465, and T442, we measured second-, third-, and fourth-order coupling terms (see Materials and Methods) and compared their magnitudes with those of off-trajectory residues. Trajectory-lining residues are defined as those residues that lie along the allosteric trajectory thought to connect the activation gate and selectivity filter elements of the channel (15). Initially, pairwise coupling was considered. As an example, the interaction between two trajectory-lining residues, A391 and E395, situated near the activation gate region, is considered. These two residues are highly coupled, as revealed from the Δ2G(i, j) value calculated from the voltage–activation curves of the four proteins comprising the double-mutant cycle [Fig. 2A and supporting information (SI) Table 1]. In line with earlier work (15) and as reaffirmed here with other trajectory-lining pairs, high coupling free energies are measured along the allosteric trajectory of the Kv channel (SI Table 2).

Fig. 2.
Pairwise coupling free energies [Δ2G(i,j)] along the allosteric trajectory of the Kv channel are abolished upon perturbation of an adjacent trajectory residue. (A) Voltage–activation curves for four channel proteins comprising the double-mutant ...

Next, the effect of a third trajectory-lining residue, k, on the second-order coupling between the trajectory-lining (i, j) residue pairs was considered [Δ3G(i,j)k]. As indicated by Horovitz and Fersht (17), a thermodynamic cube (Fig. 2B) is needed to measure the third-order coupling of this interaction. Accordingly, Δ3G(i,j)k is calculated by subtracting the free energies measured between any residue pair in the presence and absence of the native residue at a third position (front and back faces, respectively, of the cube in Fig. 2B). Returning to the representative (A391, E395) pair, coupling between these residues is dramatically diminished if either T469 or A465—i.e., the third (k) trajectory-lining residue—is mutated (Fig. 2 C and D, respectively). The same impact of trajectory-lining residue mutation on the magnitude of coupling between a given residue pair lying along the allosteric trajectory holds true for all couplings measured, except for the A391–A465 residue pair (compare black and gray bars in Fig. 2E). Interestingly, a coupling value of −1.89 ± 0.17 kcal/mol was calculated for the A391–A465 pair against the background of the E395A mutant (SI Table 2), implying that upon mutation of E395, the two residues in question interact more strongly in the open rather than the closed state. SI Table 2 presents Δ3G(i,j)k values for all measured coupled pairs.

The next layer of cooperative interactions along the energetic trajectory, Δ4G(i,j),(k,l), reflects the effect of interaction between two trajectory residues (k, l) on the magnitude of the coupling between adjacent paired trajectory residues (i, j). Δ4G(i,j),(k,l) is measured by using a four-dimensional thermodynamic construct, most easily represented as a double-mutant cycle of double-mutant cycles (17). Fig. 3A represents such a construct, as used to measure the effect of the interaction between T469 and A465 (k and l, respectively), found in the middle of the allosteric trajectory, on the magnitude of the coupling free energy between the A391, E395 pair (i and j, respectively). To calculate Δ4G(i,j),(k,l), the magnitude of the second-order coupling free energy between A391 and E395 when both T469 and A465 were mutated (lower right corner of the thermodynamic construct in Fig. 3A) was first determined. Fig. 3B shows the voltage–activation curves of the double, triple, and quadruple mutants subsequently used to calculate this coupling free energy. The results show that the coupling free energy between A391 and E395 is reduced by almost 1 kcal/mol in the T469A–A465V double-mutant background, relative to the native protein (compare Figs. 2A and and33B). This effect is not unique to the A391 and E395 trajectory pair, because (as reflected in Fig. 3C) the magnitude of five of six interactions between trajectory-lining residue pairs was significantly reduced upon elimination of neighboring interactions with other trajectory-lining pairs. Exploiting the symmetry of the four-dimensional construct, Δ4G(i,j),(k,l) for all six possible pair combinations of residues A391, E395, T469, and A465 is calculated as 4.76 ± 0.32 kcal/mol.

Fig. 3.
Pairwise coupling energies along the allosteric trajectory are significantly reduced in the absence of an adjacent interaction. (A) A four-dimensional construct (double-mutant cycle of double-mutant cycles) is used to measure the effect of the interaction ...

These results demonstrate that because coupling between any pair along the trajectory is abolished upon mutation of a third trajectory residue, and is significantly reduced in the absence of other trajectory interaction pairs, allosteric trajectory residues must be functionally essential. Hence, all coupled residues lying along the allosteric communication trajectory are important for efficient long-range coupling between distant functional elements.

Network Boundaries Are Well Defined.

To address how well defined are the boundaries of allosteric communication networks, we compared the average magnitudes of second- and third-order coupling free energies along and “normal” to the allosteric trajectory, as defined between pairs of on-trajectory residues and pairs comprising on-trajectory and adjacent off-trajectory residues, respectively (as indicated in SI Table 2 and in table 2 of ref. 15). As shown in Fig. 4, the average magnitude of pairwise coupling free energy along the allosteric trajectory (n = 12) is much stronger than for pairwise interactions “normal” to the trajectory (n = 9), despite the average distance (Å) between residue pairs in the former group being almost 2-fold higher than that in the latter (compare the 9.4 and 5 Å values given above the second-order coupling bars). As a specific example, one can consider the interactions of T469, an on-trajectory residue, with either L403 or E395, respectively, corresponding to a proximal off-trajectory and a distant on-trajectory residue, and indicated by the dashed gray and red arrows, respectively, in Fig. 1A. The coupling free energy between the T469–L403 pair is 1.23 ± 0.01 kcal/mol (15), a value 2-fold lower than the coupling free energy between the T469–E395 pair (2.55 ± 0.09 kcal/mol; see SI Table 2), despite the fact that L403 and E395 are 7 Å apart, whereas L403 and T469 are only 3.75 Å apart.

Fig. 4.
The boundaries of the allosteric trajectory are well defined. Presented here is a comparison of the averaged second- and third-order coupling free energies between trajectory-lining residue pairs and residue pairs comprising a trajectory-lining residue ...

A similar trend is observed when third-order coupling free energies [Δ3G(i,j)k] along and “normal” to the allosteric trajectory are compared (n = 5 and 3, respectively). A third residue, k, affects the magnitude of coupling between any measured trajectory-lining pair (i, j) to a much greater extent if k itself resides along the trajectory. For example, the proximal L403 off-trajectory residue affects the magnitude of coupling between the T469–A465 trajectory pair to a much lower extent than does the distant E395 on-trajectory residue [Δ3G(i,j)k of 1.35 ± 0.12 and 3.3 ± 0.14 kcal/mol, respectively]. The anisotropic direction-dependent coupling profile of trajectory-lining residues for both Δ2G(i,j) and Δ3G(i,j)k implies that the boundaries of the allosteric trajectory are well defined.

Allosteric Trajectories Are Highly Cooperative.

Efficient communication between distal functional elements would be achieved were all interactions along the allosteric trajectory highly coupled. To thus assess the degree of synergy between residues lining the allosteric trajectory, second-, third-, and fourth-order coupling free energies for six on-pathway interactions were calculated by quantifying the extent to which each pairwise interaction [Δ2G(i,j)] is affected by the presence of an adjacent trajectory-lining residue k3G(i,j)k] or interacting residue pair (k, l) [Δ4G(i,j),(k,l)] (Fig. 5A). Strikingly, for all six interaction pairs considered, Δ2G(i,j) < Δ3G(i,j)k < Δ4G(i,j),(k,l). Trajectory residue k affects the interaction between the (i, j) pair by 1 kcal/mol, on average, whereas the adjacent trajectory (k, l) interacting pair affects the interaction of the same (i, j) pair by 2.7 kcal/mol, on average (Fig. 5B). Residue-coupling along the trajectory is, therefore, highly cooperative, with the presence of an adjacent trajectory-lining residue or interacting pair progressively increasing the strength of coupling of a given trajectory residue pair. Indeed, this highlights the uniqueness of allosteric trajectory-lining residues and may explain how such residues are strongly coupled to each other over distances reaching 18 Å (Fig. 1A) (15).

Fig. 5.
Allosteric communication trajectories exhibit increasingly stronger layers of cooperative interactions. (A) Comparison of the second-, third-, and fourth-order coupling free energies associated with the indicated residue pairs along the allosteric trajectory. ...

The existence of cooperative interaction layers of increasing magnitude along the allosteric communication trajectory suggests that energy transduction along this pathway is highly coordinated. If so, then one might expect a relation between the magnitude of high-order coupling free energy along the trajectory and the Hill coefficient (nH), a measured property of cooperativity for channel gating transitions. Because the slope, Z, of voltage–activation curves of the Kv channel is directly related to nH (2224) (see SI Methods), and because ΔGopen = −FZV1/2 (where V1/2 and F are the activation midpoint and Faraday constant, respectively), it was next considered whether nH and/or V1/2 are responsible for the nonadditivity observed in high-order coupling analysis (Fig. 6). Comparison of Fig. 6 A and B reveals that both second- and third-order coupling free energies for the interactions considered in Fig. 5A are linearly correlated, with nonadditivity being associated with nH but not V1/2. Fig. 6C, summarizing the relation between the true average nth-order coupling free energy and the averaged nth-order nonadditivity associated with nH, reveals linear correlation between the two quantities.

Fig. 6.
High-order coupling free energy is related to cooperativity in channel gating. (A and B) The second- and third-order coupling free energies of the six trajectory residue pairs indicated in Fig. 5A were plotted as a function of the nonadditivity associated ...

The analysis presented in Fig. 6 assumes that Hill constants are equivalent to energies (see SI Methods for details on how nH-associated nonadditivity is calculated). Indeed, Wyman (25) showed that the Hill constant is closely related to the average free energy of interaction between a protein's binding sites (25). This assertion, combined with the analogy between the Hill coefficients for ligand binding and channel gating (24) and the fact that the linear regression curves in Fig. 6A back-extrapolate over a wide range of nH-associated nonadditivity to the (0, 0) point, suggests that the observed correlations are indeed thermodynamic in essence. Such experimental correlation also implies that high-order coupling free energies along the allosteric trajectory can be described in terms of the Hill coefficients of channel gating, in analogy to a similar theoretical finding on ligand-binding allosteric systems (26). Nonadditivity in coupling free energy may, in principal, also result from effects on V1/2 alone, or in combination with the effects of nH. The observed correlation between ΔnG and nonadditivity in nHnnH) could, therefore, not be anticipated a priori. The correlation between high-order coupling free energies and the nonadditivity associated with nH points to the cooperativity in intramolecular interactions (ΔnG) observed along the allosteric trajectory as being related to cooperativity in channel gating. In other words, the stronger the magnitude of high-order cooperative interactions along the allosteric trajectory, the more cooperative is the channel's gating transition, and vice versa. To summarize, the strong magnitude of high-order interaction free energies among trajectory-lining residues (Figs. 5 and and66C) suggests that energy transfer along trajectories of allosteric communication is highly cooperative.

Features of Long-Range Coupling in Proteins Are General.

The observed hierarchy of cooperative interactions among trajectory-lining residues contributing to channel gating requires further elaboration, in particular because no similar observations have been reported for any other protein. Insight into the significance of our findings may come from comparison with staphylococcal nuclease, the only other system where the contributions of second-, third-, and fourth-order coupling free energies to protein stability were extensively measured (2729). In the case of the nuclease, two sets of functionally and structurally different residues—namely surface-exposed and hydrophobic core residues—revealed completely different behaviors in terms of their coupling free energies. Systematic analysis of pairwise coupling free energies between distant surface-exposed residues (27) revealed the magnitude of coupling free energy among residue pairs [Δ2G(i,j)] to be linearly correlated with the nonadditivity associated with the slope value of the denaturation reaction (mGuHCl). This finding parallels what we observe in the present study of Kv channel trajectory-lining interactions. By contrast, Stites and colleagues (28, 29) concluded that packing of a protein interior can be closely approximated by a series of short-range (<6 Å), nearest-neighbor pairwise interactions [Δ2G(i,j) > 0] (28), and that second-order, but not third- or fourth-order, couplings are important for protein hydrophobic core stability [Δ3G(i,j)k = Δ4G(i,j),(k,l) = 0] (29). In other words, neither a third hydrophobic core residue, k, nor a second interacting (k, l) pair affect an adjacent (i, j) hydrophobic core interaction. Moreover, for hydrophobic core residues, no correlation between Δ2G(i,j) and the nonadditivity associated with mGuHCl was noted (28).

The striking analogy noted here, whereby distant residues contributing to the long-range coupling involved in different thermodynamic processes (i.e., channel gating and protein unfolding) in distinct proteins exhibit similar coupling profiles [Δ2G(i,j) − Δ2slope correlations], implies a general theme in long-range communication: The coupling free energy between distant residues is related to inter- or intrasubunit cooperative changes (in the case of the Kv channel and nuclease protein, respectively), possibly brought about by global conformational changes.


The high-order coupling analysis of voltage-dependent gating described here reveals features underlying allosteric communication trajectories in Kv channels. We find that all trajectory-lining residues are energetically coupled over long distances and are important for efficient energy transduction (Figs. 2E and and33C). The trajectory boundaries are, moreover, well defined, ensuring anisotropic information transfer along the trajectory pathway, rather than in other directions (Fig. 4). Trajectory-lining residues also exhibit layers of cooperative interactions of increasing magnitude (Fig. 5B), implying that all trajectory-lining residues are highly interconnected, again, to ensure efficient information conductance. The magnitude of high-order coupling along the allosteric trajectory is related to cooperativity in channel gating (nH) (Fig. 6C), meaning that the stronger the magnitude of high-order cooperative interactions along the allosteric trajectory, the more cooperative is the channel's gating transition. Together, these features may explain how cross-talk between the voltage-activated K+ channel lower activation gate and upper C-type inactivation gate, elegantly revealed by Panyi and Deutsch (12, 14), may occur.

Our findings may also offer insight into the physical mechanism underlying communication between distant channel elements. The correlation between high-order coupling free energies along the allosteric trajectory and the nonadditivity associated with nH (Fig. 6C) further implies that local and global conformational changes (indicated by ΔnG and nH, respectively) are coupled. Furthermore, the positive coupling sign associated with all measured pairwise coupling free energies along the allosteric trajectory (see Figs. 1B and and22E) implies that these interactions are stronger in the closed rather than the open state. Thus, it is possible, although not followed directly from our data, that the allosteric connectivity trajectory lining the pore of the Kv channel is a path of physical deformation along which major conformational changes leading to pore opening propagates. Upon changes in membrane potential, closed state-stabilizing interactions along the energetic trajectory break or weaken, resulting in pore opening. Comparison of the structural changes in the closed (KcsA) (30, 31) and open (MthK) (32) K+ channel crystal structures observed along the glycine gating-hinge point, controlling the disassembly of activation gate bundle crossings, is consistent with this suggestion. Although comparison of the mapping profiles of Shaker channel gating-sensitive residues on both the closed KcsA and open Kv 1.2 (33) pore structures reveals similar gross features of allosteric trajectory (not shown), closer inspection reveals subtle differences. The partners in all coupled trajectory residue pairs are always closer in the closed channel (KcsA) than in the open channel (Kv 1.2) (see SI Table 3). Taken together, our data support a mechanical view of signal transmission in which the coupling between the distal activation gate and selectivity filter elements is brought about by coordinated conformational changes transferred along an allosteric trajectory, composed of adjacent residues (4, 5, 19, 34).

The importance of the Kv channel pore allosteric communication trajectory is further emphasized by the observation that many reported Kv 1.1 and KvLQT channel pore domain mutations causing either episodic ataxia syndrome or LQT syndrome, respectively, map to gating-sensitive residues of the observed allosteric communication network (E.S., unpublished results). This finding implies that, in certain cases, the effects of disease-causing point mutations should be considered in the broader context of entire protein allosteric networks, considering that point mutation of a network residue can interfere with how distant functional element are coupled.

Our observation is striking that residues involved in long-range coupling make significant contributions to higher-order coupling terms, as observed here in the case of Kv channel allosteric trajectory-lining residues [Δ2G(i,j) < Δ3G(i,j)k < Δ4G(i,j),(k,l); see Eq. 1 and Fig. 5], whereas short-range coupling, as exemplified by hydrophobic core packing interactions, involves only residues in close proximity to each other, with no contribution from neighboring residues [i.e., Δ2G(i,j) > 0 and Δ3G(i,j)k = Δ4G(i,j),(k,l) = 0]. Whether such distinction in the interactions that residues involved in short- and long-range coupling experience is a common phenomenon requires further information on other protein systems.

Indeed, one can ask how general are the underlying features of allosteric communication networks derived here. Because the magnitude of the long-range interactions contributing to either protein stability, in the case of the staphylococcal nuclease, or channel gating, in the case of the Kv channel, is linearly related to the sensitivity (cooperativity) of the thermodynamic process to the inducing stimulus (i.e., [partial differential]ΔGunfolding/[partial differential][denaturant] and [partial differential]ΔGopening/[partial differential]V, respectively), it appears we have identified a general phenomenon. Furthermore, our data are consistent with the ideas of Ranganathan and colleagues (4, 5, 19) explaining long-range communication in proteins through pathways of energetic connectivities. Further support for this mechanical view of signal transmission comes from studies on rate–energy relationships in the acetylcholine receptor channel (35, 36). As such, the experimentally derived allosteric communication features presented here may advance the understanding of how long-range effects in proteins transpire.

Materials and Methods

High-Order Thermodynamic Coupling Analysis, and Coupling Dataset.

For descriptions, refer to SI Methods.

Molecular Biology and Electrophysiology.

Principally, all molecular biology and electrophysiology recordings techniques were performed as described in ref. 15. K+ currents were recorded from Shaker-IR channel under conditions of two-electrode voltage clamp. Voltage–activation curves were obtained by using a standard tail current protocol and represent the average of recording from 8–10 different oocytes. Care was taken to measure only oocytes expressing K+ tail currents at a narrow range of 2–4 μA. Voltage–activation curves of wild-type or mutant channels were fitted to a two-state Boltzmann equation, yielding estimates for Z (the activation slope) and V1/2 (the half-activation voltage). The free energy of channel opening, ΔGopen, at 0 mV was calculated by −FZV1/2.

Supplementary Material

Supporting Information:


We thank Profs. A. Horovitz and J. Eichler for critical reading of the manuscript. This work was supported by Israel Science Foundation Grant 323/04. O.Y. is an incumbent of the Belle and Murray Nathan Career Development Chair in Neurobiology.


The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0708120104/DC1.


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