# Sensing Magnetic Fields with Magnetosensitive Ion Channels

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Theory and Results

#### 3.1. Stochastic Dynamics without Memory

#### Separation of Closed and Open States with a Single Threshold

#### 3.2. Stochastic Dynamics in Viscoelastic Environment

#### 3.2.1. Intermediate fractional friction, ${\eta}_{\alpha}\approx {\eta}_{0}$

#### 3.2.2. Strong Fractional Friction, ${\eta}_{\alpha}\approx $ 10 ${\eta}_{0}$

## 4. Discussion

## 5. Methods

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Effective potential for the sensor rotations (in units of ${U}_{0}$), as a function of its orientation $\varphi $ (in degrees) depending on the presence of magnetic field and its orientation. The motion of sensor is restricted by the plane of membrane and the maximal angle ${\varphi}_{\mathrm{max}}\approx {34.95}^{\circ}$ due to the maximal extension length of anharmonic linker. The probability of the cluster to be in the conducting state is also depicted by the red line with symbols. For $B=0$, or for $\psi =0$, the global minimum corresponds to the closed state. In the magnetic field of Earth, the cluster is expected to be predominantly in the conducting state, e.g., for $\psi \approx {114.59}^{\circ}$ or 2 rad, where the global minimum corresponds to the open state.

**Figure 3.**Dependence of the averaged probability $\langle p(\psi ,B)\rangle $ on the angle $\psi $ (in degrees) for two values of the magnetic energy. One corresponds to ${U}_{0}=k{l}_{\mathrm{max}}^{2}$, i.e., a characteristic energy of the stretched gating springs, and another one is double of it. Notice that the sensor operation is possible already for $\mu {B}_{e}={U}_{0}$ with the maximal averaged opening probability over $0.5$. For a larger sensor with the linear sizes increased by the factor ${2}^{1/3}\approx 1.26$, i.e., $189\times 134.8\times 134.8\phantom{\rule{0.277778em}{0ex}}{\mathrm{nm}}^{3}$ (also met in living species), the maximal averaged probability increases to about $0.8$. Such a sensor would be, however, less sensitive to the variations of $\psi $ near to the maximum.

**Figure 4.**Sample trajectories (central part) and the distributions of the sensor orientation (left part), as well as distribution of current values (right part) for $\mu B={U}_{0}$ at two fixed magnetic field angles: (

**a**) $\psi =0$, with ionic channels being predominantly closed; and (

**b**) $\psi $ = 2 rad $\approx {114.59}^{\circ}$, where channels are predominantly open, in the case of Markovian memoryless dynamics. Black curves correspond to the motion of sensor and the red ones to fluctuations of ionic current due to open-closed gating dynamics. The dashed blue lines in the left parts are the corresponding theoretical values of distribution, $P(\psi ,B,\varphi )=\mathrm{exp}[-U\left(\varphi \right)/\left({k}_{B}T\right)]/\mathcal{Z}$, with $U\left(\varphi \right)$ in Figure 2. Time is in units of ${\tau}_{sc}=0.17$ ms, angle in radians, and current in the units of a maximal current possible.

**Figure 5.**Survival probabilities of open and closed times in the case of memoryless dynamics for the channels: (

**a**) predominantly closed, $\mu B={U}_{0}$, $\psi =0$; and (

**b**,

**c**) predominantly open, $\mu B={U}_{0}$, $\psi =2$. (

**a**,

**b**) The residence time distributions are extracted by placing two thresholds at the sensor orientations corresponding to the maxima of probability distribution of the sensor orientations (the left panel in Figure 4), or, equivalently, the current values (the right panel in Figure 4). (

**c**) Only one threshold is used at the minimum of current distribution corresponding to $p=0.2$ and the top of $U\left(\varphi \right)$ barrier separating two metastable states. Notice, that many re-crossings of this threshold occur when the sensor dwells on the top of this barrier. A measuring device with a finite time resolution $\Delta {t}_{res}$ will miss many of such events. We model this by using $\Delta {t}_{res}=100\delta t$, where $\delta t=2\xb7{10}^{-6}$ is the time step in simulations. Notice that this incorrect procedure leads to spurious power law and stretched exponential distributions with the parameters shown in the plot. It results also in far too small values of the mean residence times, $\langle {\tau}_{c}\rangle $, $\langle {\tau}_{o}\rangle $, as compare with the correct values in the part (

**b**). Clearly, these “measurable” mean times will become even much smaller for $\Delta {t}_{res}=\delta t$. This procedure with one threshold placed at top of the barrier separating two basins of attraction is hence very subjective and it cannot be trusted.

**Figure 6.**Sample trajectories (central part) and the distributions of the sensor orientation (left part), as well as distributions of the current values (right part) for $\mu B={U}_{0}$ at two fixed magnetic field angles: (

**a**) $\psi =0$, with ionic channels being predominantly closed; and (

**b**) $\psi $ = 2 rad $\approx {114.59}^{\circ}$, where channels are predominantly open, in the case of non-Markovian fractional dynamics with ${\eta}_{\mathrm{eff}}=1000$, ${\eta}_{\alpha}\approx 10$ ${\eta}_{0}$. Black curves correspond to the motion of sensor and the red ones to fluctuations of ionic current due to open-closed gating dynamics. The dashed blue lines in the left parts are the corresponding theoretical values of distribution, $P(\psi ,B,\varphi )=\mathrm{exp}[-U\left(\varphi \right)/\left({k}_{B}T\right)]/\mathcal{Z}$, with $U\left(\varphi \right)$ in Figure 2. Time is in units of ${\tau}_{sc}=0.17$ ms, angle in radians, and current in the units of a maximal current possible.

**Figure 7.**Survival probabilities of open and closed times in the case of non-Markovian fractional dynamics with ${\eta}_{\mathrm{eff}}=100$, ${\eta}_{\alpha}\approx {\eta}_{0}$: (

**a**) $\mu B={U}_{0}$ and $\psi =0$; and (

**b**) $\mu B={U}_{0}$ and $\psi =2$. The residence time distributions are extracted by placing two thresholds at the sensor orientations corresponding to the maxima of probability distribution.

**Figure 8.**Survival probabilities of open and closed times in the case of non-Markovian fractional dynamics with ${\eta}_{\mathrm{eff}}=1000$, ${\eta}_{\alpha}\approx 10{\eta}_{0}$: (

**a**) $\mu B={U}_{0}$ and $\psi =0$; and (

**b**) $\mu B={U}_{0}$ and $\psi =2$. The residence time distributions are extracted by placing two thresholds at the sensor orientations corresponding to the maxima of probability distribution.

© 2018 by the author. 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**

Goychuk, I.
Sensing Magnetic Fields with Magnetosensitive Ion Channels. *Sensors* **2018**, *18*, 728.
https://doi.org/10.3390/s18030728

**AMA Style**

Goychuk I.
Sensing Magnetic Fields with Magnetosensitive Ion Channels. *Sensors*. 2018; 18(3):728.
https://doi.org/10.3390/s18030728

**Chicago/Turabian Style**

Goychuk, Igor.
2018. "Sensing Magnetic Fields with Magnetosensitive Ion Channels" *Sensors* 18, no. 3: 728.
https://doi.org/10.3390/s18030728