# Information Dynamics of a Nonlinear Stochastic Nanopore System

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## Abstract

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## 1. Introduction

## 2. Definition and Estimation of Local and Specific Entropy Rates

#### 2.1. Notation

#### 2.2. Differential Entropy and Total Entropy Rate

#### 2.3. Local Entropy Rate

#### 2.4. Specific Entropy Rate

#### 2.5. Specific Information Dynamics with Python

`sidpy`) package, hosted on GitHub [40].

## 3. Model System

## 4. Results

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

LER | Local entropy rate |

SER | Specific entropy rate |

NLPL | Negative log predictive likelihood |

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**Figure 1.**Potential at fixed positive values of y (

**A**) and fixed negative values of y (

**B**). The $y=0$ configuration is shown on both plots. As y becomes more positive, it causes the potential to skew towards a transition to negative x. As y becomes more negative, it causes the potential to skew towards a transition to positive x. Physically, positive values of x in this graph correspond to positive current values.

**Figure 2.**

**Top**: the nanopore current, with each orange dot representing a measurement.

**Middle**: the estimate of the local entropy rate (LER) of the nanopore system as a function of time.

**Bottom**: the estimate of the specific entropy rate (SER) of the nanopore system as a function of time. This is a representative excerpt from a 40,000 point time series containing on the order of 100 transitions.

**Figure 3.**A projection of the reconstructed state space for the nanopore system shaded by the estimates of the LER (

**left**) and SER (

**right**) associated with the overall state. The plots reveal a clear trajectory in the reconstructed state space, and the arrows indicate the direction along the transitions between open and closed states. Along this trajectory, regions of relatively low surprise (LER) and low uncertainty (SER) occur when the system is in an open/closed state. Conversely, in the central regions, corresponding to transitions, we see increases in both the LER and SER. Anti-symmetry is noted in the onset of increase in SER, which shows that uncertainty is highest at the beginning of a transition and decreases as the transition proceeds to completion. (

**a**) LER; (

**b**) SER.

**Figure 4.**A 2D cross section of the reconstructed state space constructed from points within $\u03f5=\pm \phantom{\rule{3.33333pt}{0ex}}0.05$ of the ${x}_{t}=0$ (

**left**) and ${x}_{t-1}=0$ (

**right**) planes for each plot, respectively. These plots show that information is generated most heavily around atypical transition events (the highest LER visible on the periphery of the transition tubes in the LER plot), and there is relatively uniform, high uncertainty for all transitions in the SER plot. (

**a**) LER; (

**b**) SER.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Gilpin, C.; Darmon, D.; Siwy, Z.; Martens, C.
Information Dynamics of a Nonlinear Stochastic Nanopore System. *Entropy* **2018**, *20*, 221.
https://doi.org/10.3390/e20040221

**AMA Style**

Gilpin C, Darmon D, Siwy Z, Martens C.
Information Dynamics of a Nonlinear Stochastic Nanopore System. *Entropy*. 2018; 20(4):221.
https://doi.org/10.3390/e20040221

**Chicago/Turabian Style**

Gilpin, Claire, David Darmon, Zuzanna Siwy, and Craig Martens.
2018. "Information Dynamics of a Nonlinear Stochastic Nanopore System" *Entropy* 20, no. 4: 221.
https://doi.org/10.3390/e20040221