# AIM for Allostery: Using the Ising Model to Understand Information Processing and Transmission in Allosteric Biomolecular Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{bound}and K

_{unbound}are the equilibrium constants for the activation reactions of the receptor when bound or unbound to the allosteric ligand, respectively. An equilibrium constant can be defined in terms of concentrations or rate constants:

_{on}] and [R

_{off}] are the steady state concentrations of the receptor in the on and off state, respectively, and k

_{on}and k

_{off}are the corresponding rate constants for the transition to the on and off states (see Figure 1). The concentrations of the two receptor populations can be inferred from biochemical measurements of function, and the allosteric efficacy of the ligand of interest can be calculated from (1) and (2). When α > 1, the on state of the receptor is preferred in the presence of ligand and the ligand is considered an agonist (activator of function), and when α < 1, the off state of the receptor is preferred in the presence of ligand and the ligand is considered an inverse agonist (inhibitor of function). When α is 1, the ligand has no effect on the functional state of the receptor and the ligand is considered a neutral antagonist (inhibitor of activation by another ligand). This type of allostery, in which the equilibrium constant is modified by the ligand, is often described as “K-type”, as opposed to those that change enzyme catalysis in terms of k

_{cat}or V

_{max}, which are described as “V-type” [15].

_{x}is the power of x defined as:

^{−1}. When both the signal and noise are Gaussian, the Shannon-Hartley theorem [16,17] relates the signal-to-noise ratio to the information theoretical channel capacity C (which is the upper limit on the information rate or mutual information), by:

## 2. Results and Discussion

#### 2.1. The Thermodynamic Allosteric Efficacy as a Function of Local Interactions

_{B}T, where k

_{B}is the Boltzmann constant and T is the temperature in Kelvin. The numerator is known as the Boltzmann factor, and Z is the partition function, which sums over the Boltzmann factors of all states and normalizes the probability:

_{x}is the number of molecules of X and V is the volume, we can rewrite (2) with the explicit definition of protein concentration:

_{on}and f

_{off}are the fraction of receptors in the on and off states, respectively. Given that the system is ergodic, the frequency of a given state at steady state will converge to the ensemble probabilities. Rewriting (1) by substituting thermodynamic equilibrium constants with ratios of probabilities, we can define the allosteric efficacy as:

#### 2.2. The Allosteric Ising Model (AIM) for Multicomponent Systems

#### 2.3. Representation of allosteric propagation through specific regions within the protein

#### 2.4. The Channel as a Chain of Interacting Structural Components

#### 2.5. Comparison of Allosteric Propagation in Ising and Non-Ising Systems

^{−1}, 3/β, or 5/β. The exact allosteric efficacies, calculated from the exact probabilities of each state, were then compared to the allosteric efficacies estimated from (42) using the direct allosteric efficacy terms. We should note that while direct allosteric efficacies can be calculated for non-Ising model, the calculation of the configuration energy term followed:

#### 2.5. A Relation of AIMs to the Structural Dynamics Analysis of Biomolecular Function

#### 2.6. AIMs and Multiple Allosteric Channels

#### 2.7. Illustration of AIM-Based Analysis of Allosteric Coupling Mechanisms: The Asymmetric D2 Receptor Homodimeric Signaling Complex

^{conf}= 1), and the interaction energies between all components were negative such that they preferred to be in the same state (u

^{int}=−1). We find that this coarse grained model responds as expected to agonists, antagonists, and inverse agonists (see Figure 8B). To create a homodimer with negative cooperativity, we then added to the AIM a negative cooperativity between the one monomer that can bind G protein (which is now protomer A) and one that cannot (protomer B), represented as a positive interaction energy between their transmembrane domains (see Figure 8C). We then calculated the allosteric efficacy for the homodimer when protomer A was bound to agonist and protomer B was simultaneously bound to either an agonist, an antagonist, or an inverse agonist. This model reproduces the observed negative cooperativity (See Figure 8D)

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Shi, Y. A Glimpse of Structural Biology through X-Ray Crystallography. Cell
**2014**, 4, 995–1014. [Google Scholar] - Markwick, P.R.L.; Malliavin, T.; Nilges, M. Structural biology by NMR: Structure, dynamics, and interactions. PLoS Comput. Biol.
**2008**, 4, e1000168. [Google Scholar] - Bai, X.; McMullan, G.; Scheres, S.H. How cryo-EM is revolutionizing structural biology. Trends Biochem. Sci.
**2015**, 40, 49–57. [Google Scholar] - Ha, T. Single-molecule fluorescence resonance energy transfer. Methods
**2001**, 25, 78–86. [Google Scholar] - Sahu, I.D.; McCarrick, R.M.; Lorigan, G.A. Use of electron paramagnetic resonance to solve biochemical problems. Biochemistry
**2013**, 52, 5967–5984. [Google Scholar] - Adcock, S.A; McCammon, J.A. Molecular dynamics: Survey of methods for simulating the activity of proteins. Chem. Rev.
**2006**, 106, 1589–1615. [Google Scholar] - Bahar, I.; Lezon, T.R.; Yang, L.-W.; Eyal, E. Global dynamics of proteins: bridging between structure and function. Annu. Rev. Biophys.
**2010**, 39, 23–42. [Google Scholar] - Gether, U. Uncovering molecular mechanisms involved in activation of G protein-coupled receptors. Endocr. Rev.
**2000**, 21, 90–113. [Google Scholar] - Monod, J.; Changeux, J.-P.; Jacob, F. Allosteric proteins and cellular control systems. J. Mol. Biol.
**1963**, 6, 306–329. [Google Scholar] - Gunasekaran, K.; Ma, B.; Nussinov, R. Is allostery an intrinsic property of all dynamic proteins? Proteins
**2004**, 57, 433–43. [Google Scholar] - Makhlynets, O.V; Raymond, E.A; Korendovych, I.V. Design of allosterically regulated protein catalysts. Biochemistry
**2015**, 54, 1444–56. [Google Scholar] - Hilser, V.J.; Wrabl, J.O.; Motlagh, H.N. Structural and energetic basis of allostery. Annu. Rev. Biophys.
**2012**, 41, 585–609. [Google Scholar] - Tsai, C.; Nussinov, R. A. Unified View of “How Allostery Works.”. PLoS Comput. Biol.
**2014**, 10, e1003394. [Google Scholar] - Leff, P. The two-state model of receptor activation. Trends Pharmacol. Sci.
**1995**, 16, 89–97. [Google Scholar] - Fenton, A.W. Allostery: an illustrated definition for the “second secret of life.”. Trends Biochem. Sci.
**2008**, 33, 420–425. [Google Scholar] - Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J
**1948**, 27, 623–656. [Google Scholar] - Shannon, C.E. Communication in the presence of noise. Proc. Inst. Radio Eng.
**1949**, 37, 10–21. [Google Scholar] - LeVine, M.V.; Weinstein, H. NbIT—A New Information Theory-Based Analysis of Allosteric Mechanisms Reveals Residues that Underlie Function in the Leucine Transporter LeuT. PLoS Comput. Biol.
**2014**, 10, e1003603. [Google Scholar] - Bowman, G.R.; Geissler, P.L. Equilibrium fluctuations of a single folded protein reveal a multitude of potential cryptic allosteric sites. Proc. Natl. Acad. Sci. U.S.A
**2012**, 109, 11681–11686. [Google Scholar] - Gasper, P.; Fuglestad, B. Allosteric networks in thrombin distinguish procoagulant vs. anticoagulant activities. Proc. Natl. Acad. Sci. U.S.A
**2012**, 109, 21216–21222. [Google Scholar] - Ising, E. Beitrag zur theorie des ferromagnetismus. Zeit. Phys. A Hadron. Nucl.
**1925**, 31, 253–258. [Google Scholar] - Hopfield, J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. U.S.A
**1982**, 79, 2554–2558. [Google Scholar] - Machta, B.B.; Papanikolaou, S.; Sethna, J.P.; Veatch, S.L. Minimal model of plasma membrane heterogeneity requires coupling cortical actin to criticality. Biophys. J
**2011**, 100, 1668–1677. [Google Scholar] - Muñoz, V.; Thompson, P.A; Hofrichter, J.; Eaton, W.A. Folding dynamics and mechanism of beta-hairpin formation. Nature
**1997**, 390, 196–199. [Google Scholar] - Vorov, O.K.; Livesay, D.R.; Jacobs, D.J. Helix/coil nucleation: A local response to global demands. Biophys. J
**2009**, 97, 3000–3009. [Google Scholar] - Vorov, O.K.; Livesay, D.R.; Jacobs, D.J. Nonadditivity in conformational entropy upon molecular rigidification reveals a universal mechanism affecting folding cooperativity. Biophys. J
**2011**, 100, 1129–1138. [Google Scholar] - Bray, D.; Duke, T. Conformational spread: the propagation of allosteric states in large multiprotein complexes. Annu. Rev. Biophys. Biomol. Struct.
**2004**, 33, 53–73. [Google Scholar] - Graham, I.; Duke, T. Dynamic hysteresis in a one-dimensional Ising model: Application to allosteric proteins. Phys. Rev. E
**2005**, 71, 061923. [Google Scholar] - Perez, D.M.; Karnik, S.S. Multiple signaling states of G-protein-coupled receptors. Pharmacol. Rev.
**2005**, 57, 147–161. [Google Scholar] - Kahsai, A.W.; Xiao, K.; Rajagopal, S.; Ahn, S.; Shukla, A.K.; Sun, J.; Oas, T.G.; Lefkowitz, R.J. Multiple ligand-specific conformations of the β2-adrenergic receptor. Nat. Chem. Biol.
**2011**, 7, 692–700. [Google Scholar] - Urban, J.D.; Clarke, W.P.; Von Zastrow, M.; Nichols, D.E.; Kobilka, B.; Weinstein, H.; Javitch, J. A.; Roth, B.L.; Christopoulos, A.; Sexton, P.M.; Miller, K.J.; Spedding, M.; Mailman, R.B. Functional Selectivity and Classical Concepts of Quantitative Pharmacology. J. Pharmacol. Exp. Ther.
**2007**, 320, 1–13. [Google Scholar] - Kenakin, T. Functional selectivity and biased receptor signaling. J. Pharmacol. Exp. Ther.
**2011**, 336, 296–302. [Google Scholar] - Sethi, A.; Eargle, J.; Black, A.A; Luthey-Schulten, Z. Dynamical networks in tRNA: Protein complexes. Proc. Natl. Acad. Sci. U.S.A
**2009**, 106, 6620–6625. [Google Scholar] - Pandini, A.; Fornili, A.; Fraternali, F.; Kleinjung, J. Detection of allosteric signal transmission by information-theoretic analysis of protein dynamics. FASEB J
**2012**, 26, 868–881. [Google Scholar] - Ming, D.; Wall, M.E. Allostery in a coarse-grained model of protein dynamics. Phys. Rev. Lett.
**2005**, 95, 1–4. [Google Scholar] - Su, J.G.; Qi, L.S.; Li, C.H.; Zhu, Y.Y.; Du, H.J.; Hou, Y.X.; Hao, R.; Wang, J.H. Prediction of allosteric sites on protein surfaces with an elastic-network-model-based thermodynamic method. Phys. Rev. E
**2014**, 90, 1–10. [Google Scholar] - Witten, I.H.; Eibe, F.; Hall, M.A. Data Mining Practical Machine Learning Tools and Techniques; Morgan Kaufmann: San Francisco, CA, USA, 2005; p. 664. [Google Scholar]
- Del Sol, A.; Tsai, C.-J.; Ma, B.; Nussinov, R. The origin of allosteric functional modulation: multiple pre-existing pathways. Structure
**2009**, 17, 1042–1050. [Google Scholar] - Han, Y.; Moreira, I.S.; Urizar, E.; Weinstein, H.; Javitch, J.A. Allosteric communication between protomers of dopamine class A GPCR dimers modulates activation. Nat. Chem. Biol.
**2009**, 5, 688–695. [Google Scholar] - Moro, O.; Lameh, J.; Högger, P.; Sadée, W. Hydrophobic amino acid in the i2 loop plays a key role in receptor-G protein coupling. J. Biol. Chem.
**1993**, 268, 22273–22276. [Google Scholar] - Ballesteros, J.A.; Jensen, A.D.; Liapakis, G.; Rasmussen, S.G.F.; Shi, L.; Gether, U.; Javitch, J.A. Activation of the β2-Adrenergic Receptor Involves Disruption of an Ionic Lock between the Cytoplasmic Ends of Transmembrane Segments 3 and 6. J. Biol. Chem.
**2001**, 276, 29171–29177. [Google Scholar] - Fritze, O.; Filipek, S.; Kuksa, V.; Palczewski, K.; Hofmann, K.P.; Ernst, O.P. Role of the conserved NPxxY(x)5,6F motif in the rhodopsin ground state and during activation. Proc. Natl. Acad. Sci. U.S.A
**2003**, 100, 2290–2295. [Google Scholar] - Shan, J.; Khelashvili, G.; Mondal, S.; Mehler, E.L.; Weinstein, H. Ligand-Dependent Conformations and Dynamics of the Serotonin 5-HT 2A Receptor Determine Its Activation and Membrane-Driven Oligomerization Properties. PLoS Comput. Biol.
**2012**, 8, e1002473. [Google Scholar] - Han, D.S.; Wang, S.X.; Weinstein, H. Active state-like conformational elements in the beta2-AR and a photoactivated intermediate of rhodopsin identified by dynamic properties of GPCRs. Biochemistry
**2008**, 47, 7317–7321. [Google Scholar] - Fenwick, R.B.; Orellana, L.; Esteban-Martín, S.; Orozco, M.; Salvatella, X. Correlated motions are a fundamental property of β-sheets. Nat. Commun.
**2014**, 5, 4070. [Google Scholar] - Forrest, L.R.; Rudnick, G. The rocking bundle: a mechanism for ion-coupled solute flux by symmetrical transporters. Physiology (Bethesda)
**2009**, 24, 377–386. [Google Scholar] - Forrest, L.R.; Zhang, Y.-W.; Jacobs, M.T.; Gesmonde, J.; Xie, L.; Honig, B.H.; Rudnick, G. Mechanism for alternating access in neurotransmitter transporters. Proc. Natl. Acad. Sci. U.S.A
**2008**, 105, 10338–10343. [Google Scholar] - Nygaard, R.; Zou, Y.; Dror, R.O.; Mildorf, T.J.; Arlow, D.H.; Manglik, A.; Pan, A.C.; Liu, C.W.; Fung, J.J.; Bokoch, M.P.; Thian, F.S.; Kobilka, T.S.; Shaw, D.E.; Mueller, L.; Prosser, R.S.; Kobilka, B.K. The dynamic process of β(2)-adrenergic receptor activation. Cell
**2013**, 152, 532–542. [Google Scholar] - Provasi, D.; Artacho, M.C.; Negri, A.; Mobarec, J.C.; Filizola, M. Ligand-induced modulation of the free-energy landscape of G protein-coupled receptors explored by adaptive biasing techniques. PLoS Comput. Biol.
**2011**, 7, e1002193. [Google Scholar] - Kazmier, K.; Sharma, S.; Quick, M.; Islam, S.M.; Roux, B.; Weinstein, H.; Javitch, J.A; McHaourab, H.S. Conformational dynamics of ligand-dependent alternating access in LeuT. Nat. Struct. Mol. Biol.
**2014**, 21, 472–479. [Google Scholar] - Shi, L.; Quick, M.; Zhao, Y.; Weinstein, H.; Javitch, J.A. The mechanism of a neurotransmitter:sodium symporter–inward release of Na+ and substrate is triggered by substrate in a second binding site. Mol. Cell.
**2008**, 30, 667–677. [Google Scholar] - Zhao, C.; Stolzenberg, S.; Gracia, L.; Weinstein, H.; Noskov, S.; Shi, L. Ion-controlled conformational dynamics in the outward-open transition from an occluded state of LeuT. Biophys. J
**2012**, 103, 878–888. [Google Scholar] - Zhao, Y.; Terry, D.; Shi, L.; Weinstein, H.; Blanchard, S.C.; Javitch, J.A. Single-molecule dynamics of gating in a neurotransmitter transporter homologue. Nature
**2010**, 465, 188–193. [Google Scholar] - Zhao, Y.; Terry, D. S.; Shi, L.; Quick, M.; Weinstein, H.; Blanchard, S. C.; Javitch, J.A. Substrate-modulated gating dynamics in a Na+-coupled neurotransmitter transporter homologue. Nature
**2011**, 474, 109–113. [Google Scholar] - LeVine, M.; Perez-Aguilar, J.; Weinstein, H. N-body Information Theory (NbIT) Analysis of Rigid-Body Dynamics in Intracellular Loop 2 of the 5-HT2A Receptor, Proceedings of of International-Work Conference on Bioinformatics and Biomedical Engineering 2014 (IWWBIO 2014), Granada, Spain, 7–9 April 2014; pp. 1190–1201.

**Figure 1.**Thermodynamic cycle of a two-state ligand/receptor activation reaction. The receptor (blue circle) has an on and an off state (square and triangle indentations, respectively), both of which can bind a ligand (red triangle). The kinetic parameters are shown for the two equilibria of interest.

**Figure 2.**Schematic representations of Allosteric Ising models (AIMs). In the four AIMs analyzed here the ligand, L, is represented as a red triangle, and the protein is the blue circle subdivided into various constituent structural components. Lines separating ligand from protein or protein structural components from each other are labeled with the appropriate interaction energy term (as used in the text). Allosteric effective interactions are represented with green dotted lines. (

**A**): The simple two-component ligand/receptor system. (

**B**): A three-component ligand/receptor system with two allosteric sites, A1 and A2. (

**C**): A three-component ligand/receptor system with one channel, C, coupling the ligand and the allosteric site A. (

**D**): A four-component ligand/receptor system with two channels, C1 and C2, coupling the ligand and the allosteric site A.

**Figure 3.**The effective interaction energy through serial channels. Effective interaction energies of the first and last components of one-dimensional Ising chains are plotted as a function of chain length for conditional allosteric efficacy values of 10 (black), 100 (blue), 1000 (purple) 10,000 (red) and 100,000 (orange). The inset shows detail for short chain lengths. The effective interaction energy is seen to decay exponentially with channel length.

**Figure 4.**Using the Ising model to estimate effective interaction energies in non-Ising three-component/two-state systems. The exact effective interaction energies of 100,000 three-component/two-state non-Ising systems are plotted against the effective interaction energy estimated using the equations derived for the three-component Ising model (see(42)). The systems are generated using energy terms sampled from a normal distribution of mean 0 and standard deviation of 1/β (

**A**), 3/β (

**B**), and 5/β (

**C**) and the points are plotted with 10% opacity.

**Figure 5.**Calculated mutual information between the channel and allosteric sites sets a lower bound on the allosteric efficacy. The symmetric uncertainty between the two components is plotted against the absolute effective interaction energy for 100,000 two-component/two-state non-Ising models (

**A**), and two-component Ising models (

**B**). The systems are generated using energy terms sampled from a normal distribution of mean 0 and standard deviation of 1/β, and the points are plotted with 10% opacity.

**Figure 6.**Relation of effective interaction energies in non-Ising two-state systems with multiple independent channels to estimates from the corresponding Ising model. The exact effective interaction energies of 100,000 two-state non-Ising system is plotted against the effective interaction energy estimated using the equations derived for the n-channel Ising model (Equation (51)) for two (

**A**), and three (

**B**) independent channels. The systems are generated using energy terms sampled from a normal distribution of mean 0 and standard deviation of 1/β, and the points are plotted with 10% opacity.

**Figure 7.**The effective interaction energy of a two-channel AIM as a function of the interaction energy between the channels. (

**A**): The two-channel system in which each channel contributes to positive allosteric modulation is shown for a ligand that interacts with one channel (blue) or both channels (black). (

**B**): A two-channel system with one positive allosteric channel and one negative allosteric channel is shown for a ligand that interacts only with the positive channel (blue), only with the negative channel (red), or both channels (black). The effect of interactions between channels is seen to modify significantly the allosteric signal transduction.

**Figure 8.**Analysis of the AIM for a well-characterized asymmetric D2 homodimer of the dopamine D2 receptor (D2R). (

**A**): The D2R monomer AIM. (

**B**): The effective interaction energy calculated for the D2R monomer AIM is presented for ligands that are agonists, antagonists, and inverse agonists, and also for the mutation of either IL3 or the conserved binding motifs (CBMs). (

**C**): A molecular model of the homodimer obtained as described in the text, is shown with each AIM domain labeled in white on the structural representation. Protomer A is in blue, protomer B is in orange, and the G protein is in red. (

**D**): The effective interaction energy for the D2R homodimer AIM is presented for different combinations of the states of protomer A (indicated by

**A**in the top row) and those of protomer B in the dimer (

**B**, bottom row).

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**MDPI and ACS Style**

LeVine, M.V.; Weinstein, H. AIM for Allostery: Using the Ising Model to Understand Information Processing and Transmission in Allosteric Biomolecular Systems. *Entropy* **2015**, *17*, 2895-2918.
https://doi.org/10.3390/e17052895

**AMA Style**

LeVine MV, Weinstein H. AIM for Allostery: Using the Ising Model to Understand Information Processing and Transmission in Allosteric Biomolecular Systems. *Entropy*. 2015; 17(5):2895-2918.
https://doi.org/10.3390/e17052895

**Chicago/Turabian Style**

LeVine, Michael V., and Harel Weinstein. 2015. "AIM for Allostery: Using the Ising Model to Understand Information Processing and Transmission in Allosteric Biomolecular Systems" *Entropy* 17, no. 5: 2895-2918.
https://doi.org/10.3390/e17052895