# 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

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**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