# Ion Channel Modeling beyond State of the Art: A Comparison with a System Theory-Based Model of the Shaker-Related Voltage-Gated Potassium Channel Kv1.1

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Results

#### 2.1. Electrophysiological Experiments and Datasets

_{offset}< 45 mV, seal resistance of R

_{seal}> 200 MΩ (after whole-cell configuration), series resistance of R

_{series}< 15.5 MΩ, and membrane capacitance of C

_{slow}< 35 pF. Data were further processed based on calculated activation index (AI), maximum currents, and a subsequent manual exclusion of measurements [15].

#### 2.2. Data and Data Pre-Processing Considered for HMM and STB Model Parameterization

_{seal}< 300 MΩ and cell measurements exhibiting a high noise level or seal instabilities, resulting in a sample size of n = 60 cells for the activation curves, n = 37 cells for the deactivation curves, n = 45 cells for inactivation curves, and n = 54 cells for the ramp curves. The measured voltage steps considered for parametrization of the HMM model were limited from −50 mV to 70 mV for the activation protocol and from −80 mV to −30 mV for the deactivation protocol, representing voltage levels at which deactivation occurs after channel activation.

#### 2.3. Available HH Model and HMM of the Ion Channel Kv1.1

#### 2.4. Mathematical Concepts of Ion Channel Modelling

#### 2.4.1. The System Theory-Based Modeling Approach for the Kv1.1 Channel

_{i}(t), and intermediate output functions i

_{i}(t). The intermediate input is the output of the input nonlinear element and the input to the linear element G(s). Analogously, the intermediate output is the output of the linear element G(s) and the input of the output nonlinear element. The intermediate input and the output functions are defined in Equation (2) and Equation (3), respectively.

#### 2.4.2. The HMM-Based Kv1.1 Model

_{i}and β

_{i}represent specific gating parameters and V the applied voltage. c, d, m, k, x, and y denote rate constants without voltage-dependence. Defining ${P}_{{S}_{i}}\left(t\right)$ as the probability of being in a specific state S

_{i}at time t leads to the equation for the time evolution of the channels’ open probability ${P}_{O}\left(t\right)$ [2,40]:

_{O}, the ion channel number N

_{c}, the single channel conductance g

_{Kv}

_{1.1}, and the reversal potential E

_{K}:

_{c}was individually optimized for each measurement protocol [41]. For the given dataset, the channel number was determined as N

_{c_act}= 3088 for the measured activation and N

_{c_deact}= 2588 for the deactivation currents. The final model parameters are summarized in Table 1.

_{act_HMM}= 0.0714, RMSE

_{deact_HMM}= 0.1098). For detailed information on model parametrization and simulations, see Appendix A.

#### 2.4.3. The HH-Based Kv1.1 Model

_{1/2}denotes the half activation and inactivation voltage, k the slope factor, and A the starting point. The time constant for activation ${\tau}_{m}$ was fitted by two Boltzmann equations, and a single Boltzmann equation was again used for ${\tau}_{h}$:

#### 2.5. Evaluation, Verification, and Comparison of the Three Model Approaches

_{norm}) and absolute (RMSE

_{abs}) currents (Equation (24)).

_{max}is the maximal conductance measured at a voltage step of 70 mV, V

_{1/2_act}the hemi-activation voltage, and k

_{act}the slope factor. The activation time constant was determined by fitting a single exponential function to each individual current curve from the start of the stimulus to the peak current:

_{1/2_inact}and the slope factor of inactivation k

_{inact}were, again, calculated by fitting the normalized peak currents of the depolarizing voltage step to a Boltzmann function according to Equation (29):

_{max_cond}. Following Ranjan et al. [15], the peak value during the rising phase of the first ramp was used as the parameter V

_{max_cond}.

_{1/2_act_measured}= −22.45 mV and slope factor k

_{act_measured}= 10.81 mV were best reproduced by simulations with the HMM (V

_{1/2_act_HMM}= −22.64, k

_{act_HMM}= 11.82 mV). For the STB model, the curve and, thus, the half activation voltage were slightly shifted towards a more depolarized value, but comparable results to the HH model could be obtained with V

_{1/2_act_STB}= −18.39 and k

_{act_STB}= 14.97 mV relative to V

_{1/2_Act_HH}= −14.94 and k

_{act_HH}= 9.913 mV. With respect to the activation time constant, both the HMM and the STB model better reflected the actual voltage-dependent dynamics of activation by showing a faster activation time at higher clamp voltages and a slower activation as the voltage decreased, compared to the HH model with the same time constant over the entire voltage range. However, the activation in the STB model was instantaneous and, thus, somewhat too fast, while the activation in the HMM, especially at lower voltages, was too slow compared to the measured values. The simulation results for the deactivation of the HMM and STB models revealed a slower deactivation, but they, again, better reflected the measured deactivation behavior compared to the HH model, as shown by the determined deactivation time constants (see Figure 10 and Table 3).

## 3. Discussion

#### 3.1. Model Accuracy

#### 3.2. Model Complexity, Explainability, and Adaptability

#### 3.3. Computational Burden

#### 3.4. Experimental Data for Model Parameterization

#### 3.5. Which Method Should Now Be Chosen? When, How, and Why?

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Computational Modeling and Parameterization

**HH model.**For the HH model, differential equations of activation and inactivation gates were solved numerically by the Forward Euler method according to Ranjan et al. [15], using a step size of ∆t = 1.10

^{−4}s.

**HMM model.**The parametrization of the HMM was based on the averaged activation (n = 60) and deactivation (n = 37) data by a particle swarm optimization (PSO) algorithm (swarm size: 600; number of iterations: 10,000; function tolerance: 1 × 10

^{−6}) using the Global Optimization Toolbox (MathWorks Inc.). Defining ${P}_{{S}_{i},k}$ as the fraction of channels in a specific state S

_{i}at time-step k, the time evolution of the system could be described by the following set of autonomous difference equations:

^{−7}. The output vector was defined as ${c}^{\mathrm{T}}=\left[\begin{array}{cccccccc}0& 0& 0& 0& 1& 0& 0& 0\end{array}\right]$, to obtain the fraction of channels in the open state ${P}_{\mathrm{O},k}$ for each time-step k. The optimization defined the best choice for the voltage-dependent forward ($\alpha ,\lambda ,\sigma $) and backward state transition rates ($\beta ,\eta ,\epsilon $) and the constant state transition rates $c,d,m,k$, $x,\mathrm{and}\text{}y$ as well as the number of ion channels (${N}_{{C}_{Kv1.1}}$) by fitting the resulting macroscopic (${I}_{\mathrm{model}})$ current to the measured whole-cell current (${I}_{\mathrm{data}})$:

**STB model.**The main advantage of this approach is that the model equations (see Equation (7)) do not have to be solved numerically. Instead, the identified model can be exported into the workspace of MATLAB where the obtained model can be further analyzed, linearized, or inserted into Simulink for a further application and simulation. The dynamic behavior of the ion channel is finally characterized by the transfer function and input/output nonlinearities.

## Appendix B. Simulation of AP and Recovery Voltage Protocols with the HMM and STB Models

**AP:**Action potential protocol consisted of a 1.8 s long regular spiking action potential at a 30 mV voltage and frequency of 18 Hz, mimicking the physiological stimuli of pyramidal neurons.

**Recovery:**Recovery characteristics from inactivation were measured with 16 recovery pulses of a 200 ms duration at 50 mV after channel inactivation by 1.5 s long pre-pulses at 50 mV, and the recovery of the cells by holding them at −80 mV for variable times between 50 ms and 3.2 s in steps of 150 ms.

**Figure A1.**Simulation of additional (

**a**) action potential (AP) and (

**d**) recovery protocols with the HMM (

**b**,

**e**) and STB models (

**c**,

**f**). Corresponding RMSE values for simulation of the AP protocol were RMSE

_{AP_HMM}= 0.0435 and RMSE

_{AP_STB}= 0.0321; for the recovery protocol: RMSE

_{recovery_HMM}= 0.1200 and RMSE

_{recovery_STB}= 0.0831. RMSE values excluding the depolarization pulses of recovery protocols calculated from 1.5 s to the end of the test pulses were RMSE

_{recovery_HMM}= 0.0777 and RMSE

_{recovery_STB}= 0.0945. Ion channel numbers N

_{c}used for simulation of the macroscopic current with the HMM model were (

**b**) N

_{c_AP}= 2900 and (

**e**) N

_{c_recovery}= 2600.

## Appendix C. Additional Simulation of the Ramp Curve with the STB Model

**Figure A2.**Optimization result of the STB model using 10 breakpoints for stepwise linearization of the non-linear input and output function. (

**a**) Absolute ramp currents (RMSE

_{ramp_STB_abs}= 0.0023) and (

**b**) normalized ramp currents (RMSE

_{ramp_STB_norm}= 0.0013). Black: measured current data; green: simulated current data.

## Appendix D. Original Hodgkin–Huxley Formalism of the Potassium Current

_{x}and driving force (V − E

_{x}) of the ion:

_{x}is described by various gates, controlling the flow of ions through the membrane. Each gate contains several independent gating particles z, which change between the open and closed positions, depending on the membrane potential. The gating variable y represents the probability of a single gating particle being in the open state. For several independent gating particles z, the probability of the entire gate being open, is given by y

^{z}.

**Figure A3.**Transition reaction. C: closed state, O: open state, α: voltage-dependent forward transition, β: voltage-dependent backward transition.

_{0}is the starting point at time zero, ${y}_{\infty}$ the steady state value, and ${\tau}_{y}$ the time constant. Both, ${y}_{\infty}$ and ${\tau}_{y}$ are related to the voltage dependent rate coefficients α(V) and β(V), which can further be modeled by fitting empirical functions of the membrane potential to experimental data:

_{K}:

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**Figure 1.**Voltage step and ramp protocols. (

**a**) Activation protocol, (

**b**) deactivation protocol, (

**c**) inactivation protocol, and (

**d**) ramp protocol.

**Figure 4.**Results of the Kv1.1. STB model identification: (

**a**) input voltage data using the ramp protocol in mV, output current data in nA representing the measured Kv1.1 macroscopic current; (

**b**) optimization result of the STB model. Black: measured current data; green: simulated current data (RMSE

_{ramp_STB}= 0.0364).

**Figure 5.**Alpha-subunit of the shaker-related voltage-gated potassium channel Kv1.1; VSD: voltage sensor domain; PD: pore domain. Created with BioRender.

**Figure 6.**HMM of the Kv1.1 channel; C: closed; O: open; IC: slow inactivation; IN: fast inactivation.

**Figure 7.**Optimization result of the HMM for (

**a**) activation data (−90 mV to 70 mV, ∆V = 10 mV, RMSE

_{act_HMM}= 0.0714) and (

**b**) deactivation data with zoomed deactivation pulses (−80 mV to −30 mV, ∆V = 10 mV, RMSE

_{deact_HMM}= 0.1098). Black: measured current data; blue: simulated current data. The first line represents the respective voltage step protocols.

**Figure 8.**Fractional occupancy plot of hidden Markov model simulations for (

**a**) activation protocol at voltage level of 50 mV and (

**b**) deactivation protocol at voltage level of −50 mV. C: closed state; O: open state; IC: slow inactivation states; IN: fast inactivation states. The first line represents the respective voltage step protocols.

**Figure 9.**Model simulations of (

**a**–

**c**) activation protocols (−90 to 70 mV, ∆V = 10 mV), (

**d**–

**f**) deactivation protocols with zoomed deactivation pulses (−80 to −30 mV, ∆V = 10 mV), (

**g**–

**i**) inactivation protocols (−80 to −70 mV, ∆V = 10 mV), and (

**j**–

**l**) ramp protocols by the HH, HMM, and system theory-based approaches, respectively. Ion channel numbers N

_{c}used for simulation of the macroscopic current with the HMM model are (

**b**) N

_{c_act}= 3088, (

**e**) N

_{c_deact}= 2588, (

**h**) N

_{c_inact}= 2588, and (

**k**) N

_{c_ramp}= 2388. Voltage step and ramp protocols for simulations are shown in Figure 1.

**Figure 10.**Measured (black) and simulated electrophysiological parameters by the HH (red), HMM (blue), and STB (green) models. (

**a**) Conductance plot of activation, (

**b**) conductance plot of inactivation, (

**c**) time constant of activation, (

**d**) time constant of deactivation, and (

**e**) time constant of inactivation.

Rate Constants and Parameters | |||||
---|---|---|---|---|---|

α_{1} | 951.2464 s^{−1} | λ_{1} | 14.1140 s^{−1} | σ_{1} | 3.8031 s^{−1} |

α_{2} | 0.03 V | λ_{2} | 20.2499 V | σ_{2} | 11.8850 V |

β_{1} | 395.7896 s^{−1} | η_{1} | 49.9528 s^{−1} | ε_{1} | 58.364 s^{−1} |

β_{2} | 0.0501 V | η_{2} | 5 V | ε_{2} | 55.3568 V |

c | 799,720 s^{−1} | k | 370.9594 s^{−1} | x | 1.6056 s^{−1} |

d | 38,916 s^{−1} | m | 1199.6 s^{−1} | y | 0.0822 s^{−1} |

E_{K} | −0.065 V | g_{Kv}_{1.1} | 8.7 pS | ||

N_{c_act} | 3088 | N_{c_deact} | 2588 |

HH | HMM | STB | |
---|---|---|---|

unknown parameters | 22 | 20 | 7 |

mathematical description of the model | 2 first-order differential equations | 8 first-order differential equations | 1 third-order differential equations |

parameterization data | activation single-cell measurements | activation and deactivation average | ramp average |

number of cells | 56 | 60 (activation) 37 (deactivation) | 54 |

voltage range | −40 to +50 mV | −90 to +70 mV (activation) −80 to −30 mV (deactivation) | −80 to +70 mV |

total sweep numberconsidered | 10 | 13: −50 to +70 mV (activation) 6: −80 to −30 mV (deactivation) | 1 |

time for parameterization/system identification | data not available | 30 h | 5–10 min |

Temp 35 °C | Experimental Data | Simulated Data | |||
---|---|---|---|---|---|

HH | HMM | STA | |||

activation | |||||

V_{1/2_act} | (mV) | −22.45 | −14.94 | −22.64 | −18.39 |

k_{act} | (mV) | 10.81 | 9.913 | 11.82 | 14.97 |

${\tau}_{act\_mean}$ | (ms) | 0.5493 | 0.5766 | 2.2449 | 0.2706 |

${\tau}_{act\_70mV}$ | (ms) | 0.09283 | 0.3036 | 0.1391 | 0.01875 |

${\tau}_{act\_60mV}$ | (ms) | 0.1135 | 0.2949 | 0.2125 | 0.02084 |

${\tau}_{act\_50mV}$ | (ms) | 0.1403 | 0.2865 | 0.3157 | 0.02339 |

${\tau}_{act\_40mV}$ | (ms) | 0.1791 | 0.2783 | 0.4613 | 0.02634 |

${\tau}_{act\_30mV}$ | (ms) | 0.2351 | 0.2703 | 0.668 | 0.02994 |

${\tau}_{act\_20mV}$ | (ms) | 0.3148 | 0.2629 | 0.9654 | 0.03359 |

${\tau}_{act\_10mV}$ | (ms) | 0.4343 | 0.2567 | 1.401 | 0.06175 |

${\tau}_{act\_0mV}$ | (ms) | 0.6244 | 0.2524 | 2.052 | 0.1491 |

${\tau}_{act\_-10mV}$ | (ms) | 0.9504 | 0.2486 | 3.033 | 0.3984 |

${\tau}_{act\_-20mV}$ | (ms) | 1.476 | 0.2430 | 4.437 | 0.8607 |

${\tau}_{act\_-30mV}$ | (ms) | 2.02 | 0.2374 | 6.077 | 1.613 |

RMSE_{norm}RMSE_{abs} | 0.0326 - | 0.0213 0.0714 * | 0.0138 0.0381 | ||

deactivation | |||||

${\tau}_{deact\_mean}$ | (ms) | 13.3627 | 18.5689 | 5.0230 | 10.76 |

${\tau}_{deact\_-30mV}$ | (ms) | 23.42 | 0.1704 | 3.236 | 14.86 |

${\tau}_{deact\_-40mV}$ | (ms) | 16.75 | 3.433 | 5.282 | - |

${\tau}_{deact\_-50mV}$ | (ms) | 11.49 | 31.03 | 6.491 | - |

${\tau}_{deact\_-60mV}$ | (ms) | 10.79 | 26.07 | 6.058 | 4.793 |

${\tau}_{deact\_-70mV}$ | (ms) | 7.306 | 25.68 | 5.019 | 4.564 |

${\tau}_{deact\_-80mV}$ | (ms) | 10.42 | 25.03 | 4.052 | 18.82 |

RMSE_{norm}RMSE_{abs} | 0.0429 - | 0.0627 0.1098 * | 0.0283 0.0985 | ||

inactivation | |||||

V_{1/2_inact} | (mV) | −26.46 | −29.12 | −28.95 | −27.37 |

k_{inact} | (mV) | 4.755 | 3.882 | 5.04 | 4.074 |

${\tau}_{inact\_mean}$ | (ms) | 102.1077 | 71.9092 | 96.4150 | 99.1621 |

${\tau}_{inact\_70mV}$ | (ms) | 53.22 | 32.14 | 68.15 | 53.2 |

${\tau}_{inact\_60mV}$ | (ms) | 63.13 | 32.14 | 68.45 | 60.65 |

${\tau}_{inact\_50mV}$ | (ms) | 69.17 | 32.17 | 68.85 | 68.74 |

${\tau}_{inact\_40mV}$ | (ms) | 72.65 | 32.25 | 69.41 | 71.38 |

${\tau}_{inact\_30mV}$ | (ms) | 77.25 | 32.57 | 70.25 | 77.26 |

${\tau}_{inact\_20mV}$ | (ms) | 80.26 | 33.79 | 71.6 | 85.05 |

${\tau}_{inact\_10mV}$ | (ms) | 85.9 | 38.27 | 73.89 | 83.26 |

${\tau}_{inact\_0mV}$ | (ms) | 104.3 | 53.18 | 78.19 | 101.5 |

${\tau}_{inact\_-10mV}$ | (ms) | 147.1 | 89.9 | 87.19 | 143.1 |

${\tau}_{inact\_-20mV}$ | (ms) | 208.7 | 138.4 | 107.7 | 192.4 |

${\tau}_{inact\_-30mV}$ | (ms) | 263.1 | 168.4 | 153.8 | 252.6 |

${\tau}_{inact\_-40mV}$ | (ms) | 0.5125 | 179.7 | 239.5 | 0.8051 |

RMSE_{norm}RMSE_{abs} | 0.0257 - | 0.0548 0.1297 * | 0.0146 0.0463 | ||

ramp | |||||

V_{max_cond} | (mV) | 69.6 | 67.0 | 69.2 | 69.6 |

RMSE_{norm}RMSE_{abs} | 0.1098 - | 0.0396 0.0317 * | 0.0262 0.0364 |

_{c}used for simulation of the macroscopic current with the HMM model and calculation of RMSE

_{abs}values were for activation N

_{c_act}= 3088, deactivation N

_{c_deact}= 2588, inactivation N

_{c_inact}= 2588, and ramp N

_{c_ramp}= 2388.

HH | HMM | STB | |
---|---|---|---|

explainability of channel gating | + | +++ | n.a. |

flexibility and adaptability | + | +++ | + |

model complexity | + | +++ | + |

model accuracy | (<<) + | ++ (>>) | +++ |

comp. burden optimization | ++ | +++ | + |

comp. burden simulation | + | ++ | + |

experimental data for model parameterization | +++ | +++ (>>) | + |

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**MDPI and ACS Style**

Langthaler, S.; Lozanović Šajić, J.; Rienmüller, T.; Weinberg, S.H.; Baumgartner, C.
Ion Channel Modeling beyond State of the Art: A Comparison with a System Theory-Based Model of the Shaker-Related Voltage-Gated Potassium Channel Kv1.1. *Cells* **2022**, *11*, 239.
https://doi.org/10.3390/cells11020239

**AMA Style**

Langthaler S, Lozanović Šajić J, Rienmüller T, Weinberg SH, Baumgartner C.
Ion Channel Modeling beyond State of the Art: A Comparison with a System Theory-Based Model of the Shaker-Related Voltage-Gated Potassium Channel Kv1.1. *Cells*. 2022; 11(2):239.
https://doi.org/10.3390/cells11020239

**Chicago/Turabian Style**

Langthaler, Sonja, Jasmina Lozanović Šajić, Theresa Rienmüller, Seth H. Weinberg, and Christian Baumgartner.
2022. "Ion Channel Modeling beyond State of the Art: A Comparison with a System Theory-Based Model of the Shaker-Related Voltage-Gated Potassium Channel Kv1.1" *Cells* 11, no. 2: 239.
https://doi.org/10.3390/cells11020239