GPCR Signaling: A Study of the Interplay between Structure, Energy, and Function
RUNNING TITLE : the Role of Energy Dissipation in GPCR Signaling

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Yann Chalopin¹, *

¹Structures, Properties and Modeling of Solids Laboratory
Physics Department

CentraleSupélec/National Center for the Scientific Research,
University of Paris-Saclay, 91190 Gif-sur-Yvette, France

(Dated: April 26, 2024)

G-protein coupled receptors (GPCRs) exemplify sophisticated allosteric communication, transduc-
ing extracellular signals through ligand-induced structural rearrangements that resonate through the
molecular scaffold. Despite extensive study, the biophysical underpinnings of how conformational
changes spread remain unclear. This work employs a novel physics-based framework to charac-
terize the role of energy dissipation in directing intramolecular signaling pathways. By modeling
each residue as a network of coupled oscillators, we generate a localization landscape depicting
the vibrational energy distribution throughout the protein scaffold. Quantifying directional energy
flux between residues reveals distinct pathways for energy and information transfer, illuminating
sequences of allosteric communication. Our analysis of CB1 and CCR5 crystal structures unveils an
anisotropic pattern of energy dissipation aligning with key functional dynamics, such as activation-
related conformational changes. These anisotropic patterns of vibrational energy flow constitute
pre-configured channels for allosteric signaling. Elucidating the relationship between structural
topology and energy dissipation patterns provides key insights into the thermodynamic drivers of
conformational signaling. This methodology significantly advances our mechanistic understanding
of allostery in GPCRs and presents a broadly applicable approach for rationally dissecting allosteric
communication pathways, with potential implications for structure-based drug design targeting these
critical receptors.
In the dynamic landscape of molecular biology, proteins emerge as sophisticated orchestrators of cellular functions, 10 with allostery playing a pivotal role in their regulatory mechanisms. In the field of proteins, the term *allostery* 11 commonly refers to any form of intramolecular communication that occurs through specific networks of amino acids, 12 allowing structural information to be transmitted between distant parts of the protein. While significant progress 13 has been made in understanding large-amplitude movements involving major protein domains, the subtler aspects of 14 allostery remain elusive. These involve the binding of small ligands that modify the conformational and dynamical 15 properties of distant active sites without inducing dramatic topological changes. Our project aims to shed new light 16 on this unresolved issue by establishing hierarchical networks of energetic pathways that connect distant regions of 17 macromolecules. We will focus on fast timescales (typically on the order of picoseconds), which are ideally suited for 18 rapid and efficient information transmission. 19

The earlier works of Monod et al. underscored the significance of thermal equilibrium in determining protein conformation populations$^{1-3}$, while Koshland's insights focused on the role of binding interactions in driving conformational changes$^{4-6}$. Later, experimental studies detected residues involved in allostery by comparing wild and mutated proteins. Based on such an approach, long-range mechanical coupling was found to be facilitated by disordered structures when they exist in proteins, a process deeply influenced by topological irregularities$^{7-9}$. On the other hand, functional connections between distant residues often involve independent modules or domains, termed 'sectors', within the protein$^{10-13}$. Models employing Green's functions$^{14}$ have illuminated the complex dynamics between force propagation and collective protein behavior. Lastly, evolutionary studies$^{15,16}$ have underscored a paradigm shift in protein adaptation, revealing that proteins can retain their allosteric functions despite amino acid variations, as long as their overall structure is preserved. This concept aligns with our guiding idea that if proteins consist of numerous variable residues surrounding a core of a few conserved residues, the plasticity of this particular composition enables consistent performance in allosteric interactions amidst diverse internal and external stresses$^{15,16}$. In the realm of allostery, traditional studies have focused on pinpointing crucial residues through point mutations, normal modes analysis, and molecular dynamics simulations$^{17-20}$. Molecular dynamics and Normal Modes Analysis methods capture dynamical correlations between secondary structures or domains but are generally applied to only the wild type and perhaps one or two mutants of the protein of interest. While these approaches offer qualitative insights, they often fail to identify expansive networks of amino acids that span binding pockets. Moreover, these methods focus on the lowest-energy, largest-amplitude, and slowest motions, which represent the fraction of a protein's energy spectrum (or density of states). On the other hand, experimental mutational studies have provided potential insights into residues involved in allostery but are constrained by time and resource limitations. While often involving small molecules like metabolites or signaling molecules, proteins can also act as effectors in these processes. However, the identification of networks that could ensure the communication between distant functional sites is still lacking. In fact, the impact of rapid timescale events in allosteric phenomena, particularly those in the picosecond range, remains unexplored. 21

This study aims to explore and build upon these seminal works by further exploring the potential role of thermal fluctuations in protein allostery. More specifically, we investigate how thermal agitation induces vibrations within the protein scaffold, which depends on the structural configuration, and how this property enables proteins to dynamically respond to environmental stimuli. For that, we focus on G protein-coupled receptors (GPCRs), whose functions are based on long-range effects of various ligands. These well-known proteins also enable us to shed light on the evolutionary pressures for functional discrimination in protein structures, a concept we have thoroughly explored and reported in our previous research$^{21}$.

Our research focuses on the interplay between the structure and energy dissipation of receptor intramolecular communication for two well-characterized systems: CCR5 and CB1. The C-C chemokine receptor type 5 (CCR5) is a G protein-coupled receptor expressed on leukocytes that plays a vital role in immune cell trafficking by binding chemokine ligands$^{22}$. As a major co-receptor for HIV-1, CCR5 is also an important antiviral drug target for preventing viral entry into host cells$^{23}$. Both orthosteric and allosteric binding sites allow various small molecule ligands to activate CCR5 through conformational changes that initiate intracellular signaling cascades$^{24}$. Understanding the precise structural mechanisms underlying CCR5 activation will inform future drug development efforts.

Similarly, the cannabinoid receptor type 1 (CB1) is a G protein-coupled receptor abundantly expressed in the central and peripheral nervous systems that regulates diverse physiological processes, including pain, mood, appetite, and cognition$^{25}$. Both endogenous cannabinoids like anandamide and exogenous ligands such as $\Delta^9$-tetrahydrocannabinol (THC), the primary psychoactive constituent of cannabis, can activate CB1$^{26}$. Elucidating the intricacies of CB1 activation and downstream signaling will provide molecular insights instrumental for developing targeted CB1-based therapies for chronic pain, neurological disorders, metabolic diseases, and more$^{27}$. In summary, detailed understanding of the structural mechanisms of CCR5 and CB1 activation by various ligands is key for enabling structure-based drug design.
Using GPCR proteins as basis model, this study proposes that intramolecular communication entails information transfer among distant residues, extending beyond the interactions established through direct contacts. We aim to quantify this process using the concept of entropy exchange between residues²⁸. While it is well established that proteins exhibit energy and entropy variations during transitions between conformational states via Smoluchowski diffusion, our focus is on the exchange of information between residues in native structures: once proteins achieve conformational stability, the regular spacing (~3.8Å) between backbone residues allows for the emergence of thermally activated atomic movements known as phonons*. These lattice vibrations serve as energy carriers and play a crucial role in dissipation processes. Upon mechanical perturbations, phonons transport heat flux, leading to local entropy creation and reduction*. However, energy transport in proteins differs from solid crystals due to the irregular molecular topology resulting from folding, which leads to wave localization. This phenomenon, where energy is confined to specific regions due to structural heterogeneity, requires central consideration in our analysis as it gives rise to dynamically decoupled domains within the protein (referred to as dynamic clusters*), essential prerequisites for the selective long-range control of distinct groups of residues. A basic assumption in our work is that the interactions between residues can be approximated harmonically. This is appropriate provided that the displacements from equilibrium positions remain small, which we will quantitatively confirm later. Therefore, for the purpose of mathematical derivation, the potential energy of residues approximates a quadratic form of displacements from equilibrium positions $x_i$ as

$$V = \frac{1}{2} \sum_{i,j} \mathbb{H}_{i,j} x_i x_j + \mathcal{O}\|x\|^3. \quad (1)$$

Where $\mathbb{H}$ stands for the force constant matrix. For each degree of freedom $i$ interacting with neighboring residues $j$ of mass $m_\alpha$, $\alpha \in i, j$ the equation of motion writes,

$$\ddot{x}_i = - \sum_{j \neq i} \frac{\mathbb{H}_{i,j}}{m_i} x_i + \sum_{j \neq i} \frac{\mathbb{H}_{i,j}}{\sqrt{m_i m_j}} x_j \quad (2)$$

Equ. 2 describes the collective dynamics of each amino acid's oscillating from their equilibrium position, in the harmonic regime ($x_i = y_i e^{i\omega t}$), such that it is written in the form of an eigenvalue problem:

$$\tilde{\mathbb{H}}\mathbf{Y}^n = \omega_n^2 \mathbf{Y}^n \quad (3)$$

$\omega_n^2$ being the eigenvalues of $\tilde{\mathbb{H}} = (\mathbb{H}_{i,j})$, corresponding to the square of the vibrations pulsation. In this study, we utilize a parameter-free coarse-grained elastic network models (pfENMs) to produce the dynamical matrices $\tilde{\mathbb{H}}^{29}$. For the purpose of this work, we assume here that ENMs offer accurate enough predictions of protein normal modes³⁰.

Dynamic matrices function as elliptic operators, allowing us to compute a "localization landscape" ($u_h$) to predict where waves are likely to localize within a system³¹ depending their energy or frequency.

$$-\mathbb{H}_{\pi} u_h(x) = (1)$$

Here, (1) stands as a unit vector, $\mathbb{H}_\pi$ is the dynamic matrix projected at the Brillouin zone's edge, accounting for the periodic nature of the protein backbone with a wavelength $\lambda = 3.8$ and wave vector $k = \pi/\lambda$.

$$\mathbb{H}_{\pi}^{ij} = \begin{cases} c - \mathbb{H}_{i,j}, & \text{if } i = j, \\ \mathbb{H}_{i,j} \times (-1)^{|i-j|}, & \text{if } i \neq j. \end{cases}$$

Here, $c$ is a constant greater than the highest eigenvalue of $\tilde{\mathbb{H}}$ to ensure $u_h$ is positive. $|i-j|$ represents the distance between residues $i$ and $j$. From $u_h$, we derive an "effective confining potential"³² $W(x)$, computed as $W(x) = 1/u_h(x)$. The vibration frequency $\nu$ is then determined by $\nu = \sqrt{W-c}/2\pi$. This potential informs us how waves are confined: high values create barriers against wave penetration, while low values create "wells" that trap waves. These contrasting behaviors lead to localized "wave packets" or dynamic clusters within the system. Previous research has shown that dynamic clusters with the fastest vibrations are associated with catalytic activity in proteins²¹. These clusters are regions where local compression between residues occurs, affecting the energy barriers essential for catalysis. This helps to explain the phenomenon known as rate-promoting vibrations (RPVs)³³,³⁴. The localization landscape $u_h$ has a direct relationship with the system's phonon response, thanks to its correspondence with the static limit of the Green's function¹⁴. This finding extends our understanding of how localized vibrations, or phonons, enable
information transfer between residues involved in chemical regulation. We introduce a new metric, the "entropy flux for localized thermal phonons," to elaborate on this idea. It provides insight into the mechanisms governing protein dynamics and regulation. Since energy dissipation is necessary for information transfer³⁵, calculating the heat carried by these localized vibrations is crucial for understanding this complex system. Assuming harmonic atomic interactions, vibrational modes exhibit no lifetime or mean free path, leading to heat transfers that are governed by a coherent transport regime. The heat flux $q_T^{ij}$ between two different points, i and j, in the system can be calculated using the eigenvectors of the dynamic matrix $\mathbb{H}^{36}$. The formula for this is:

$$q_T^{ij} = k_B(T_j - T_i) \sum_n \frac{|Y_i^n|^2 \cdot |Y_j^n|^2}{|Y_i^n|^2 + |Y_j^n|^2} \quad (4)$$

$$= k_B(T_i - T_j)\mathbb{T}_{ij} \quad (5)$$

Here, $T_i$ and $T_j$ represent the temperatures of residues i and j, respectively. These temperatures differ when the system experiences a local mechanical disturbance, pushing it away from equilibrium. From the second principle of thermodynamics, the entropy flux* is obtained from the heat flux divided by the temperature.

$$q_S^{ij} = \frac{q_T^{ij}}{T_0} \quad (6)$$

$$= k_B \delta_T \mathbb{T}_{ij} \quad (7)$$

A practical approach for many engineering applications is to use an average or an effective temperature. This could be the arithmetic mean, $T_0 = (T_i + T_j)/2$, or a logarithmic mean if the temperatures differ significantly. The quantity $\mathbb{T}_{ij}$ is a 2D tensor mapping of how each degrees of freedom (or residue) exchanges information with the rest of the structure. The total entropy associated with each degree of freedom is then obtained from the relation

$$\Sigma_S^i = \sum_{j \neq i} q_S^{ij} \quad (8)$$

$$A_i = \sum_{j \neq i} q_S^{ij} r_{ij} \quad (9)$$

## A. Structures

## RESULTS AND DISCUSSION ON CCR5

<footnote n="94">⁹⁴ We assume that the system is slightly perturbed so that $\delta_T = \frac{T_i - T_j}{T_i} \ll 1$. Since intramolecular communication</footnote>

<footnote n="95">⁹⁵ corresponds to long-range communication, we evaluate the allosteric distance score*, which is defined by the amount</footnote>

<footnote n="96">⁹⁶ of transmitted information weighted by the Cartesian distance separating residues or groups of residues.</footnote>

<footnote n="97">⁹⁷ In summary, Our approach couples each residue with a variable number of degrees of freedom through an inho-</footnote>

<footnote n="98">⁹⁸ mogeneous equation of motion (eq. 2). This gives rise to a localization landscape (eq. ) that characterizes the</footnote>

<footnote n="99">⁹⁹ vibrational energy distribution across the protein structure. Based on this landscape, we quantify the anisotropic flux</footnote>

<footnote n="100">¹⁰⁰ of vibrational energy between residues (eq. 5). This enables us to identify pathways for long-range communication,</footnote>

<footnote n="101">¹⁰¹ whereby distant residues can effectively exchange entropy. To account for distance effects, we balance the entropy</footnote>

<footnote n="102">¹⁰² exchange efficiency between a source and receiving residue against the spatial separation between them (eq. 9).</footnote>

<footnote n="104">¹⁰⁴ The Protein Data Bank (PDB) structures 5XRA and 4MBS serve as invaluable resources for our study. Structure</footnote>

<footnote n="105">¹⁰⁵ 5XRA, with a resolution of 2.8 Å, provides a detailed snapshot of the active state of G-protein-coupled receptor</footnote>

<footnote n="106">¹⁰⁶ (GPCR). Conversely, 4MBS, resolved at 3.1 Å, captures the GPCR inactive conformation. Together, they provide the</footnote>

<footnote n="107">¹⁰⁷ essential basic structural framework for investigating the behavior of GPCRs, enabling the dissection of the directional</footnote>

<footnote n="108">¹⁰⁸ flux of energy and intramolecular communication within GPCRs in two relevant distinct states.</footnote>

<footnote n="110">¹¹⁰ As a chemokine receptor, CCR5 belongs to the expansive class A family of G protein-coupled receptors (GPCRs),</footnote>

<footnote n="111">¹¹¹ which accounts for over 80% of all GPCRs and includes many therapeutically relevant targets such as adrenergic,</footnote>

<footnote n="112">¹¹² dopaminergic, and opioid receptors³⁷. Members of this class share common structural motifs like seven transmem-</footnote>

<footnote n="113">¹¹³ brane helices and conserved sequences within the transmembrane bundle³⁸. Studies have revealed similarities in the</footnote>
<figure><img src="image_1.png" /><figcaption>FIG. 1: **Structural Analysis and Thermal Vibration Localization in CCR5 Receptor.** This figure presents a comprehensive view of the CCR5 receptor's crystallographic structure (PDB ID: 4mbs) and its thermal energy distribution, detailed in three panels: (a) Demonstrates the confinement potential in relation to the full spectrum of normal modes, highlighting energy distribution over various timescales. (b) Focuses on the hydrophobic pocket (in blue), the binding site for drugs like Maraviroc. (c) Features a color-coded vibrational map, categorizing the protein into two dynamic clusters with rapid (red) and slow (blue) vibrations.</figcaption></figure>

114 dynamic conformational changes associated with class A GPCR activation, suggesting potential generalizability of key 115 mechanistic principles³⁹. However, there is still considerable diversity among class A receptors in their ligand binding 116 sites, signaling mechanisms, and allosteric modulation⁴⁰. While insights from CCR5 may inform our understanding 117 of activation for other class A GPCRs, further structure-function analyses would be required to extend these findings 118 to specific receptors beyond CCR5.

119 We specifically investigate the network of residues at both the Maraviroc binding pocket and the gp120 binding 120 interface in CCR5 (Figure 1a). To this aim, we first introduce a mathematical tool termed the confinement potential 121 *W* that allows for a spectral decomposition of the protein structure into domains defined by their characteristic time 122 dynamics (See Materials and Methods). This approach links each domain to its dynamic behavior or characteristic 123 timescale, enabling a detailed mapping of the protein's topological landscape to its dynamic attributes. Figure 1b
<figure><img src="image_2.png" /></figure>

FIG. 2: **Anisotropic Information flux and Vibrational Frequencies in CCR5.** This figure shows the interplay between the CCR5 receptor's tertiary structure (PDB ID: 4mbs) and its vibrational dynamics. (a) Displays an entropy map for residue pairs calculated using Eq. 10. (b) Illustrates the total entropy flux (Eq. 11), overlaid on the 3D structure, highlighting residues (in red) with a high communication potential. (c) Depicts allosteric scores (Eq. 12) in relation to residue conservation frequency, identifying areas of extensive long-range communication involving less conserved residues.

124 visualizes CCR5's dynamics through a 3D color-coded model, where each color corresponds to domains as defined by the aforementioned decomposition. From this analysis, three distinct dynamic clusters within CCR5 emerge. The 125 slower-frequency domain, represented in blue and operating below 1.0 THz, aligns with the extracellular (EC) and 126 intracellular (IC) regions. Conversely, the high-frequency domain, highlighted in red at around 3 THz, is primarily 127 associated with the transmembrane (TM) region. Notably, the Maraviroc binding pocket is centrally located at 128 the interface between the extracellular (EC) and transmembrane (TM) domains, as will be demonstrated. This 129 preliminary dynamic segmentation closely corresponds to the primary structural architecture of the CCR5 receptor 130 consisting of EC, TM, and intracellular (IC) domains. 131

132 Having revealed distinct dynamical domains, we must elucidate how these modules exchange energy and information. 133 Selective coupling likely enables specific communication pathways that facilitate allosteric control. Decoding this 134 modular energetic network will provide insight into functional mechanisms and opportunities for rational modulation. 135 Quantifying information flow between the dynamically decoupled residues is central to unraveling allosteric regulation. 136 To identify key residues that enable communication, we quantified the capacity for entropy exchange between all 137 residue pairs using Equation 7 (Figure 2a). This analysis, grounded in the second law of thermodynamics whereby 138 entropy flow tracks accessible states, revealed primary conduits for information propagation based on vibrational 139 energy transfer. 140

140 In this study, the term *entropy flux* refers to the heat flux carried by localized thermal phonons divided by temperature, as defined by the second law of thermodynamics. This definition allows us to quantify the flux of entropy between residues in the protein, providing insights into the mechanisms governing protein dynamics and regulation. 141 142
143 It is important to note that this definition of entropy flux differs from the concept of entropy flux quantified by local 144 Lyapunov exponents, which measure the rate of separation of infinitesimally close trajectories in phase space and are 145 not the focus of this work. 146

146 Residues exhibiting high communication capacity are postulated to facilitate allosteric signaling by propagating
147 mechanical fluctuations that disrupt local (covalent or non-covalent) interactions. A preliminary visualization of the
148 residue communication capacity projected onto the CCR5 tertiary structure suggested the lateral transmembrane
149 helices may facilitate signal transduction across the lipid bilayer. Nevertheless further investigation is warranted to
150 finely characterize their role as allosteric pathways (Figure 2b).

151 To further elucidate long-range communication pathways across the entire CCR5 structure, we computed an 'al-
152 losteric score' for each residue by weighing its communication capacity against distance, as described in the Materials
153 and Methods. This allosteric score metric identifies residues capable of transmitting information over long ranges,
154 revealing potential conduits for allosteric signaling, as shown in Figure 2b.

155 Analysis of the full vibrational density of states reveals peak conformational entropy exchange occurs at 1.5 THz, 156 corresponding to the densest spectral region with the highest eigenmode count. This indicates vibrational energy 157 transfer is optimized when mediated through the collective in-phase motions of residues coupled to abundant proximal 158 oscillators operating at resonant frequencies. Information transmission relies on this fine balance - sufficiently focused 159 spectral density to enable coherent dynamics yet broad enough to prevent localization. Further elucidating the 160 relationship between the protein's dynamic conversation and underlying topology remains vital for decoding the 161 grammar of allostery.

162 Our findings suggest a potential trend of residues with higher vibrational frequencies exhibiting reduced communica-
163 tion potential, though exceptions likely exist. For example, the dynamic cluster from residues index 230 (corresponding
164 to residue number N252) to 250 (S272), part of the external loop EL3, exhibits slower vibrations alongside a constrained
165 transport capacity. In contrast, the cluster around residue index 183 (G202 at the TM5 interface) combines rapid
166 vibrations with substantial entropic flux. Notably, high-frequency residues capable of effective communication usually
167 operate within limited domains, whereas those with lower frequencies facilitate broader long-range transport. Fur-
168 thermore, we found that the frequency range most effective for long-range information transmission aligns with the
169 findings of Rouviere41 highlighting the importance of collective (and thus slower) motions in facilitating long-range
170 communication within proteins. Interestingly, we also noted that 'soft domains', positioned in the middle range of
171 stiffness, are the most efficient in signal transmittance within the protein systems.

172 This frequency-localization relationship corroborates Leitner et al., who identified domain spatial extent as key
173 for elucidating vibrational control of communication pathways42-44. Together, these findings underscore the critical
174 interplay between structural dynamics, energetic properties, and efficient functional signaling in proteins. We hypoth-
175 esize that this relationship is that high-frequency protein regions exhibit greater stiffness and higher cohesive energy,
176 characterized by elevated force constants that enable efficient energy transmission. These high-frequency networks
177 typically involve a smaller number of residues, resulting in more localized domains. In contrast, lower-frequency or
178 softer domains span across larger regions within the molecule, encompassing a greater number of residues. While indi-
179 vidual couplings in these low-frequency domains may be weaker, the collective integration of these numerous couplings
180 can still result in significant energy flux. In the context of our coarse-grained model, high frequencies are considered
181 to be around 3 THz, while low frequencies are typically below 1 THz. The mid-range frequencies, particularly around
182 1.5 THz, correspond to the band with the highest density of states, where the number of modes is maximal. As
183 previously demonstrated, this frequency range is optimal for entropy exchange, as it allows for the greatest number
184 of residues to share the same characteristic timescale, facilitating effective communication within the protein.

185 In addition, we examined highly conserved residues as characterized in earlier studies⁴⁵. While these exhibit high
186 structural stability based on their high-frequency timescales (Figures 2c), they showed limited communication capac-
187 ity. Integrating evolutionary studies¹⁵,¹⁶ with our current findings suggests an intriguing paradigm: residues under
188 significant mutational pressure have adapted greater versatility in intramolecular communication while retaining the
189 structural stability necessary for function under diverse conditions. This balance between conserving structural in-
190 tegrity yet enabling signaling adaptability may have been crucial over evolutionary timescales. Rather than mere
191 randomness, our findings raise the intriguing possibility of a potential role for intrinsic structural disorder in facil-
192 itating communication across dynamic residue clusters. This disorder allows targeted signaling through frequency
193 channels densely populated with communicative nodes, as evident in Figure 2C. The resulting expansion of potential
194 allosteric sites confers an evolutionary advantage by enhancing modulation of distant interactions. The nuanced in-
195 terplay between structural dynamics and functional regulation seems to be accessible by decoding the thermodynamic
196 principles governing this signaling fluidity.

197 Our localization landscape analysis suggests that rather than behaving as a wholly interconnected network where all
198 residues are energetically coupled, proteins seem to contain specific regions that are locally decoupled from the overall
199 system. This localized decoupling enables targeted functional fluctuations to occur within certain areas, providing
200 pockets of flexibility while maintaining stability of the overall scaffold. The regional isolation allows residues in
201 these domains to undergo conformational changes independently, facilitating specific signaling motions or binding
202 interactions. At the same time, global structural integrity is preserved since the fluctuations are confined to discrete
203 regions instead of propagating across the entire network. This strategic balance between local flexibility and global
204 stability may be integral to the protein's ability to undergo functional conformational changes in response to stimuli
205 while avoiding widespread destabilizing effects.

206 This regional decoupling of controlled structural domains map distinct timescales onto specialized functional purposes. The low and high extremes of the frequency spectrum are intricately linked to critical biological processes. 207 Lower frequencies generally correspond to activating conformational changes$^{46,47}$, whereas higher frequencies are more 208 often related to rate-promoting vibrations (RPVs) at catalytic sites$^{21}$. 209

210 An interesting question emerges regarding the introduction of sufficient disorder to yield functional irregularities 211 within the cohesive stability of the folded state. Is there a critical protein size threshold required to incorporate 212 adequate decoupled regions? This adds a new dimension to comprehending protein dynamics, implying that increasing 213 complexity during evolution is intricately linked to growing molecular size. The global-local tradeoffs enabling discreet 214 plasticity zones to emerge within an otherwise rigid architecture likely become more viable above a certain length scale. 215 Further elucidating the thermodynamic principles and minimum requirements for controlled functional fluctuations 216 to arise presents an intriguing avenue for future exploration. 217

218 The findings suggest that proteins may have evolved to utilize a range of vibrational frequencies, potentially reflecting 219 diverse functional roles within the protein structure. Further research is needed to explore this hypothesis across 220 different protein families.

221 Our identification of allosteric communication pathways in CCR5 based on THz vibrational energy flux builds upon 222 and complement recent studies characterizing energy transport networks in other GPCRs. Recent works$^{48}$ and$^{49,50}$ 223 compared distinct energy flux patterns between inactive and active states of the β2-adrenergic receptor. These works 224 propose a link between efficient energy conduction and allosteric regulation in GPCRs. Our study puts forth the 225 hypothesis that energy flux facilitates entropy flux, enabling propagation of mechanical fluctuations that modulate 226 distal residue contacts. Thermally-driven atomic displacements alter or disrupt such distal interactions, whether 227 covalent or noncovalent. This highlights the potential role of phonons—the primary energy carriers—in dissipation 228 for functional ends.

229 This proposed coupling between energy flow and targeted yet stochastic signaling presents a unique structural 230 attribute that may prove fundamental to effector interactions. Let's dive into that question :

231 The interaction between HIV-1's gp120 glycoproteins and the human CCR5 receptor is crucial for viral entry. More- 232 over, in contact with gp120 (Figure 3-h), residues forming the CCR5 interface are notably conductive, underscoring 233 the critical role of vibrational dynamics in protein contact-induced conformational changes. Key residues, namely 234 Y37, W86, W248, Y251, and Q283, constitute the primary binding interface for this interaction (19,20,51). Adjacent 235 to this interface lies a hydrophobic pocket, accommodating various ligands. Our computational analysis reveals a 236 network of coupled residues at both the binding interface and the hydrophobic pocket (Figure 3b). This network 237 facilitates long-range intramolecular communication within CCR5 (Figure 3c), pointing to intricate interplays within 238 the receptor.

239 A particularly intriguing finding is that proximity does not necessarily confer dynamical coupling, as evidenced 240 by certain residues (e.g K170) spatially located near key functional sites yet displaying no energetic connectivity 241 (Figure 3). This selectivity reveals that vibrational energy transfer within the protein architecture follows discrete 242 pathways rather than a diffusive network. The underlying structural mechanisms that encode these targeted allosteric 243 communication routes, restricting the participation of some proximal residues while enabling long-range signaling 244 by distant residues, warrant further elucidation. The ability for a residue to influence binding pocket fluctuations 245 depends on more than just spatial location. As seen here, K170 donot engages a significant communication with 246 residues integral to the gp120 binding interface (Figure 3b). This selective and position-dependent coupling provides 247 empirical support for our theory: it suggests a nuanced mechanism through which proteins might modulate binding 248 affinity, especially at downstream sites in the extracellular domain.

249 This also suggests that almost any residue could potentially act as a pivot point for distal/long range communication 250 effect, underscoring the receptor's intricate and versatile communication network. Such functional adaptability enables 251 the system to dynamically respond to a diverse array of stimuli. Moreover, changes in the protein's topology, dictated 252 by its sequence, have the potential to alter these complex couplings. This flexibility affords nature a broad canvas to 253 engineer novel interfaces, which can exert influence over remote regions within the protein's structure. This capability 254 illustrate the potential adaptability and evolutionary ingenuity inherent in protein design.

255 This raises an interesting question for further investigation: Does protein size influence the presence and extent 256 of decoupled regions within the structure?: Does the observed data suggest an intrinsic cross-talk between the ex- 257 tracellular and intracellular domains of proteins, orchestrated by specific structural motifs and allosteric residues. 258 This cross-talk, facilitating information transfer over distances up to 4 nm within the protein, hints at a complex 259 internal communication system. Considering the importance of protein sectors, it becomes imperative to investigate 260 whether these motifs are fundamental building blocks, selected for their distinct vibrational properties. The presence 261 of anisotropic transport within proteins not only supports this hypothesis but also raises intriguing possibilities about 262 the evolution of sophisticated intramolecular regulatory mechanisms, including allosteric and orthosteric effects. In 263 the context of understanding anisotropic energy flow and its implications in allosteric transitions, several studies have
<figure><img src="image_3.png" /><figcaption>FIG. 3: **Functional Communication Pathways in CCR5.** This figure provides an in-depth look at the CCR5 receptor (PDB ID: 4mbs) and its internal communication networks. (a)-(d) Show entropy flux calculations for critical residues, particularly at the gp120 interface, with graphical representations highlighting allosteric channels. Arrows indicate regions of high reactivity, with red dots marking Maraviroc binding sites and their influence on the gp120 interface. (e) Presents a 3D view of the overall entropy flux of gp120-contacting residues (PDB ID: 6meo). (f) Details the entropy profile of residue K170, showing limited communication with nearby interface residues.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>FIG. 4: **Transmembrane Communication Pathways in a CB1 Membrane Receptor.** (a) Visualizes communication pathways extending over 5nm from the intracellular to the extracellular parts of the receptor, highlighting extensive transmembrane communication. (b) Focuses on the network of communicating residues within the intracellular region, particularly those coupled with the Toggle Switch mechanism. (c) Illustrates how the extracellular loop interacts with other protein regions, demonstrating unique anisotropic communication properties.</figcaption></figure>

263 investigated the intricate signaling pathways involved, including the works by$^{52–56}$

## B. Limits of our model and perspectives

While our elastic network analysis provides valuable insights into intrinsic allosteric communication pathways in CCR5 based on harmonic fluctuations, the equilibrium model is intrinsically limited in capturing the diversity of non-covalent interactions and chemical forces critical for allostery. For instance, hydrogen bonds, salt bridges, and disulfide bonds are not finely accounted for and may significantly modulate residue couplings. Furthermore, the potential contribution of water-mediated interactions to mediating conformational transitions and long-range signaling cannot be incorporated within this theoretical framework. Extensive studies on class A, B, and C GPCRs indicate that ordered water molecules can form bridging networks involved in allosteric signaling and stabilizing distinct receptor conformations through hydrogen bonding$^{57,58}$. A more comprehensive understanding of the molecular determinants underlying CCR5 allostery will require elucidating the cooperative interplay between intrinsic protein dynamics and water-mediated couplings using enhanced molecular dynamics simulations with explicit water models. This could reveal additional communication routes and stabilizing interactions supplemental to the pathways identified through our harmonic approximation. Integrating water dynamics represents an important direction for future investigations to fully capture the multifaceted nature of CCR5 allostery.
# I. RESULT AND DISCUSSION ON CB1

The type 1 cannabinoid receptor (CB1) is intricately structured, featuring an orthosteric site for endogenous ligand interaction and an allosteric site that modulates the receptor's activity. Particularly expressed in the central nervous system, CB1 plays crucial roles in various biological functions. Ligand binding at the allosteric site influences the receptor's response to the orthosteric ligand. Our analysis focused on the interactions within CB1 in the presence of THC, CBD, and CP55940, as well as the receptor's functional domains⁵⁹,⁶⁰. CB1 comprises two main residue types: those involved in molecular binding, including the allosteric and orthosteric sites, and those triggering conformational changes, known as "switches." Notably, the Toggle Switch (TS) domain within CB1, which vibrates at approximately 3 THz, is essential for receptor activation and exhibits dynamic coupling with distant residues (Figure 4). This region acts as a hinge, facilitating the conformational shifts necessary for signal transduction and illustrating a complex mechanism of intramolecular communication within the receptor. To accurately interpret information flux between domains of varying sizes ($N_i$ residues) within regions of interest ($D_i$), we normalize the fluxes as follows:

$$J_{AB} = \frac{1}{N_A} \sum_{i \in D_A} \frac{1}{N_B} \sum_{j \in D_B} J_S^{ij}, \qquad (10)$$

$$\tilde{J}_{AB} = \frac{1}{N_A} \sum_{i \in D_A} \sum_{j \in D_B} \|J_S^{ij}\|_{\infty}. \qquad (11)$$

Here, $J$ represents the normalized flux used to identify large clusters of communicating residues (Figure 5), while $\tilde{J}$ represents a similar metric that identifies clusters with communication properties achieved through a very small number of highly conducting residues. $J$ highlights the most energetically significant regions for coupling, and $\tilde{J}$ identifies smaller dynamic clusters that contain only a few highly effective residues for coupling. Our analysis of the CB1 receptor structure identifies three primary domains engaging in significant communication (Figures 4 and 5): the G protein interface (Gprot), the allosteric site (CBD), and the external loop (EC loop). Despite their spatial separation (panning from 1 nm to nearly 5 nm), these regions exhibit dynamic coupling, especially across the intramembrane interface, with the allosteric pocket forming strong connections to both extracellular and intracellular domains. Further analysis reveals that the binding site of the synthetic molecule CP55940 also demonstrates considerable dynamic coupling with these key domains (Figure 5). Using the $\tilde{J}$ metric, we pinpointed three smaller yet critical domains: the TS in the transmembrane section (highlighting the importance of this domain in receptor activation), the G protein interface (indicating its role in signal transduction), and the allosteric site (emphasizing its role in modulating receptor activity). These smaller domains, despite their size, play a crucial role in the overall function of the CB1 receptor, demonstrating the intricate and complex nature of intramolecular communication within the receptor. Our observations lend credence to existing receptor activation models in the literature¹⁸,⁶¹⁻⁶³. We observed a pronounced coupling between the Toggle Switch (TS) domain and the orthosteric site (THC). Additionally, the TS domain interacts with a series of internal activation mechanisms, notably internal switch 1 and the rotameric trigger, oriented towards the intracellular interface. This intricate network of interactions facilitates a cascade of reactions in response to vibrational perturbations, crucial for the receptor's conformational changes and activation. While our study reveals a discernible, albeit not dominant, distal interaction between the allosteric and orthosteric sites (Figure 5c), it is important to note that the resolution limitations of the extracellular (EC) loop might affect this observation's precision. Nonetheless, the allosteric site notably influences the EC loop, particularly the transmembrane (TM) interface, modulating the receptor's response to CBD. This interaction with the TM interface, which is also connected to the EC loop, illustrates the allosteric site's significant role in modulating the receptor, distinct from the primary agonist THC's effects¹⁷,¹⁸. This complex pathway of interactions, spanning from the receptor's external to internal regions, supports a model of receptor activation through dynamic perturbations⁶⁰,⁶⁴,⁶⁵. Our analysis suggests that THC binding at the orthosteric site triggers a sequence involving the toggle switch, rotameric trigger, internal switches SW1 and SW2, leading to the activation of the ionic lock. While intermediate structures between the receptor's active and inactive states remain elusive, these findings strongly imply a sophisticated molecular cascade underlying receptor activation.

# II. COMPARING ACTIVE VERSUS INACTIVE STATE

Our analysis of CB1 molecular communication reveals distinct allosteric networks in the inactive and active receptor states (Figure 6). In the inactive conformation, enhanced coupling is observed between the ionic lock and G protein interface. This supports existing models where the ionic lock sequesters G protein coupling in the basal state⁶⁶,⁶⁷. Similarly, increased connectivity between the orthosteric anandamide binding site and ionic lock likely contributes additional stability⁶⁶,⁶⁷. Concurrently, the TM2-TM3 interface and extracellular loop couple strongly, restricting movements associated with activation⁶⁸,⁶⁹.
<figure><img src="image_5.png" /><figcaption>FIG. 5: **Information flux in a CB1 membrane inactivated receptor.** (a) Provides a detailed nomenclature of the various domains involved in protein activation. (b) Showcases the entropy profile between two domains. The black curve represents communication along the sequence coordinates of the Toggle Switch, with red dots indicating THC binding residues. This method can be replicated for each pair of domains mentioned in (a). (c) Presents a matrix representation of the communication between the referenced domains of CB1 (PDB 5xra). The matrix uses both $J_D$ (right) and $\tilde{J}_D$ (left) to reveal networks of coupled domains, highlighting their varying sizes and interactions.</figcaption></figure>

<footnote n="315">315 Conversely, the activated CB1 state displays marked remodeling of allosteric pathways. The extracellular loop 316 exhibits increased crosstalk with the orthosteric THC and CBD pockets, enabling agonist-induced structural 317 rearrangements$^{68,69}$. THC binding also couples strongly with the TM3-TM4 toggle switch, a key microswitch 318 for stabilization of active GPCRs$^{68,70}$. Indeed, enhanced communication is seen between all ligand binding sites – 319 THC, CBD, anandamide and CP55940 – with the G protein interface, facilitating downstream signaling in the active 320 conformation$^{68,70}$. Connectivity between CBD and the TM5-TM6 interface may provide additional stability$^{68,69}$.</footnote>

<footnote n="322">322 Taken together, these findings provide alternative insights into the nuanced modulation of CB1 allostery. Our characterization of entropic drivers underpinning communication network remodeling substantiates molecular mechanisms 323 critical for functional transitions. The inactive state maintains firm connectivity between regulatory motifs like the 324 ionic lock and toggle switch to constrain signaling. Activation then possibly requires breaking these contacts to 325 allow agonist-bound conformations with expanded allosteric pathways and effector coupling. By mapping this flux of 326</footnote>
<figure><img src="image_6.png" /><figcaption>FIG. 6: **Remodeling of CB1 Allosteric Communication Networks Upon Activation** Changes in coupling between key structural regions of CB1 in inactive versus active states. Red color indicates the higher degree of communication, representing stronger coupling. In the inactive state, connectivity is enhanced between the ionic lock, anandamide binding site, TM2-TM3 interface, and extracellular loop. The active state displays increased communication between the extracellular loop, orthosteric ligand binding pockets, toggle switch, TM5-TM6 interface, and G protein coupling domain. This remodeling of allosteric pathways underpins the transition between basal and activated CB1 conformations.</figcaption></figure>

<footnote n="327">327 signals between specific microdomains, our thermodynamic framework enriches understanding of CB1 signaling and</footnote>

<footnote n="328">328 drug targeting.</footnote>
CONCLUSION

330 This study elucidates a biophysical basis for the nuanced propagation of allosteric signals within GPCRs, shedding 331 light on the complex conformational pathways underpinning their versatile functionality. By modeling the protein as a 332 network of coupled oscillators and quantifying the landscape of energy localization, we unveil the anisotropic patterns 333 along which thermal fluctuations dissipate through the structural scaffold. The directed routes of vibrational energy 334 flow manifest as intrinsic channels for allosteric communication, aligning with functionally relevant conformational 335 transitions. Our analysis transcends previous conceptions of mechanical coupling or generic thermal noise, revealing 336 how the topology of energy dissipation encodes signaling pathways within the receptor architecture. The breakthrough 337 afforded by this physics-based framework is the ability to trace directional information flow with residue-level precision, 338 systematically mapping sequences of allosteric communication. Extending this approach to ligand-bound states would 339 provide further insight into how binding events modulate energy localization and inter-residue transfer to induce 340 downstream structural rearrangements. By relating the geometry of dissipation to functional dynamics, we can 341 potentially decode nature's design principles for how GPCRs balance stability and plasticity. From an evolutionary 342 perspective, the conservation of an anisotropic yet malleable dissipation landscape may constitute a fundamental 343 requirement in adapting this protein family to diverse signaling tasks. Overall, this research significantly advances 344 understanding of the thermodynamic forces guiding allosteric communication within GPCRs. The methodology 345 introduces a versatile toolset to rationally map pathways of conformational signaling based on energy flow. In a 346 broader context, this work lays the foundation to elucidate general physical principles of dynamic allostery applicable 347 across molecular biophysics. Insights gained here open exciting opportunities for researchers to design and fine-tune 348 novel allosteric modulators to selectively target specific receptor conformations and signaling behaviors.
349

ACKNOWLEDGMENTS

350 YC is supported by a public grant overseen by the French National Research Agency (ANR) as part of the "In-
351 vestissements d'Avenir" program (Labex NanoSaclay, reference: ANR-10-LABX-0035). YC would like to thank Dr.
352 M. Buckle and Brigitte Hartmann for insightful discussions and edits. YC would also like to thank F. Rey for taking
353 the time to read and provide suggestions for this paper.

AUTHOR CONTRIBUTIONS

DECLARATION OF INTEREST

356 The author confirms that there are no known conflicts of interest associated with this publication.

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