# Intrinsic Computation of a Monod-Wyman-Changeux Molecule

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Causal States $\mathcal{S}$, Statistical Complexity ${C}_{\mu}$, the $\u03f5$-Machine, and the Mixed-State Simplex

#### 2.2. Monod-Wyman-Changeux Molecules

## 3. Results

#### 3.1. Intrinsic Computation

#### 3.2. Functional Computation

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A dynamical single-site Monod-Wyman-Changeux molecule, with kinetic rates as shown. States marked ${A}_{i}$ are active with i bound ligand molecules, while states marked ${I}_{i}$ are inactive with i bound ligand molecules. When transitioning from states ${A}_{0},\phantom{\rule{3.33333pt}{0ex}}{A}_{1}$, A is emitted, while when transitioning from states ${I}_{0},\phantom{\rule{3.33333pt}{0ex}}{I}_{1}$, I is emitted.

**Figure 2.**Box-counting dimension of the mixed-state presentation changes drastically with ${f}_{T}^{\prime},\phantom{\rule{3.33333pt}{0ex}}{b}_{T}^{\prime}$. For both processes, we have: ${f}_{T}=1.0,\phantom{\rule{3.33333pt}{0ex}}{f}_{A}c=2.9,\phantom{\rule{3.33333pt}{0ex}}{b}_{A}=3.4,\phantom{\rule{3.33333pt}{0ex}}{b}_{T}=3,\phantom{\rule{3.33333pt}{0ex}}{f}_{I}c=4,\phantom{\rule{3.33333pt}{0ex}}{b}_{I}=2$. The process with nonzero ${f}_{T}^{\prime},\phantom{\rule{3.33333pt}{0ex}}{b}_{T}^{\prime}$ has a scaling of $log{N}_{\u03f5}\sim log(1/\u03f5)$ and thus a nonzero box-counting dimension ${h}_{0}>0$, whereas the process with ${f}_{T}^{\prime}={b}_{T}^{\prime}=0$ has a scaling of $log{N}_{\u03f5}\sim loglog(1/\u03f5)$ and thus a box-counting dimension ${h}_{0}=0$.

**Figure 3.**At left, a generative model of the process generated by the MWC molecule in fixed ligand concentration c of Figure 1 with ${f}_{T}^{\prime}={b}_{T}^{\prime}=0$. The dwell time distributions ${\varphi}_{A}\left(t\right)$ and ${\varphi}_{I}\left(t\right)$ are given in Equations (8) and (9). At right, the corresponding topological $\u03f5$-machine. While emitting A, one moves along the “conveyer belt” starting with state A to the left; while emitting I, one moves along the conveyer belt starting with state I to the right. To switch the letter that one is emitting, one jumps to the other conveyer belt. The states along the conveyer belt to the left correspond to the time that one has been inactive, and the states along the conveyer belt to the right correspond to the time that one has been active.

**Figure 4.**${\varphi}_{A}\left(t\right)$ for ${f}_{A}={b}_{A}=1.0$ and ${f}_{T}=1.0$ (blue), ${f}_{T}=2.0$ (orange), and ${f}_{T}=3.0$ (green), calculated using Equation (8).

**Figure 5.**Contour plot of ${C}_{\mu}$ (

**a**) and $\mathbf{E}$ (

**b**) as a function of ${f}_{T},\phantom{\rule{3.33333pt}{0ex}}{b}_{T}$ when ${f}_{T}^{\prime}={b}_{T}^{\prime}=0,\phantom{\rule{3.33333pt}{0ex}}{f}_{A}c={f}_{I}c={b}_{A}={b}_{I}=1$.

**Figure 6.**Probability of being in the active state, ${p}_{eq,A}\left(c\right)$, as a function of ligand concentration c, for ${f}_{T}=1,\phantom{\rule{3.33333pt}{0ex}}{f}_{A}=100,\phantom{\rule{3.33333pt}{0ex}}{b}_{A}=0.1$ and ${b}_{T}=1,\phantom{\rule{3.33333pt}{0ex}}{f}_{I}=1,\phantom{\rule{3.33333pt}{0ex}}{b}_{I}=1$, and: ${f}_{T}^{\prime}={b}_{T}^{\prime}=0$ (blue); ${f}_{T}^{\prime}=0.01$ and ${b}_{T}^{\prime}=0$ (orange); and ${f}_{T}^{\prime}=0$ and ${b}_{T}^{\prime}=10$ (green), almost overlaying the blue.

**Figure 7.**Transfer function $G\left(\omega \right)$ as a function of input frequency $\omega $ at randomly chosen initial concentration ${c}_{0}$ for ${f}_{T}=1,\phantom{\rule{3.33333pt}{0ex}}{f}_{A}=100,\phantom{\rule{3.33333pt}{0ex}}{b}_{A}=0.1$ and ${b}_{T}=1,\phantom{\rule{3.33333pt}{0ex}}{f}_{I}=1,\phantom{\rule{3.33333pt}{0ex}}{b}_{I}=1$, and: ${f}_{T}^{\prime}={b}_{T}^{\prime}=0$ (blue); ${f}_{T}^{\prime}=0.01$ and ${b}_{T}^{\prime}=0$ (orange); and ${f}_{T}^{\prime}=0$ and ${b}_{T}^{\prime}=10$ (green), almost overlaying the blue.

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Marzen, S. Intrinsic Computation of a Monod-Wyman-Changeux Molecule. *Entropy* **2018**, *20*, 599.
https://doi.org/10.3390/e20080599

**AMA Style**

Marzen S. Intrinsic Computation of a Monod-Wyman-Changeux Molecule. *Entropy*. 2018; 20(8):599.
https://doi.org/10.3390/e20080599

**Chicago/Turabian Style**

Marzen, Sarah. 2018. "Intrinsic Computation of a Monod-Wyman-Changeux Molecule" *Entropy* 20, no. 8: 599.
https://doi.org/10.3390/e20080599