**4. Discussion**

#### *4.1. System Analysis*

The mathematical model of the system is given by Equations (3) and (4) in the Laplace domain. The unit step response of the system is shown in Figure 6. Besides the gain and time constant, other essential characteristics of the system are the rise time, transient time, settling time, and steady-state. These system characteristics are calculated from the transfer function or are found by plotting the step response in MATLAB.

**Figure 6.** Unit step response of the system *G*(*s*) with confidence region.

In Figure 6, the rise time is 0.104 s, the transient time is 0.184 s, and the final value corresponds to a gain of 0.174 nA; thus, the system does not have an overshoot. Experimental output data and the mathematical model indicate that stability is enforced. The same conclusion was drawn by conducting a pole location analysis. The model has one pole with a negative real part, and the system was determined to be stable [22,24,25].

The model in the Laplace domain corresponds to the differential equation under the assumption that all initial conditions are zero [22]. The inverse Laplace transformation gives a solution of a differential equation in the time domain, and the system described with the solution to Equation (4) in the time domain is

$$\log(t) = 0.1738 \left( 1 - e^{\frac{t}{0.0777}} \right) h(t) \tag{5}$$

where *g*(*t*) corresponds to *G*(*s*), representing the unit step response of the system *G*(*s*) (see Figure 6) when all initial conditions are equal to zero, and *h*(*t*) is the input function, which corresponds to the unit step or Heaviside function *h*(*t*).

We simulated outputs of the model using Equation (4) with inactivation and deactivation stimuli. The compared results of the measured outputs and simulated outputs are shown in Figure 7. The model satisfies the dynamic behavior of the Kv10.1 voltage-gated ion channel based on the adopted assumptions.

Since different cellular mechanisms can affect the opening behavior of ion channels, we believe that time-dependent transfer functions can be used to provide a more accurate description of the dynamic characteristics of ion channels. This hypothesis is supported by the fact that all mathematical systems are non-stationary and that the mathematical origin of non-stationarity can be proved [31].

#### *4.2. Control Algorithm*

The model (Equation (3)) is stable, observable, and controllable because the rank of the controllability and observability matrix is equal to *n*, where *n* = 1, and the system is a first-order system [22,24,25]. We considered a classical control algorithm for the Kv10.1 model. The controller is designed using the Sisotool in MATLAB. We chose an automated tuning method for a PID compensator in a feedback loop, the tuning method is a robust time response, and the controller type is PID. The controller is given by

$$C(s) = 316.85 \left( \frac{1 + 0.0026s}{s} \right) \tag{6}$$

**Figure 7.** Measured and simulated output macroscopic current for the inactivation and deactivation input stimuli (grey lines). Input stimulus inactivation: measured output (**a**), simulated output (**b**); input stimulus deactivation: measured output (**c**), simulated output (**d**).

The unit step response of the controlled Kv10.1 voltage-gated ion channel in a closedloop, also called a feedback loop, is shown in Figure 8. The Kv10.1 in the feedback loop displays a rise time of 0.043 s, a peak amplitude of 1.06 nA, an overshoot of 6.08%, a transient time of 0.149 s, and a final value of 1 nA. The controlled model has a faster rise time and transient response, and the final value corresponds to the desired value.

**Figure 8.** Unit step response of the Kv10.1 model in a control loop.

The feedback control system with the Kv10.1 model, *G*(*s*), and controller *C*(*s*) in a loop was implemented in Simulink (Figure 9a). A sine waveform with unit amplitude and frequency of 1 Hz was chosen as a simulation input. The simulated output is shown in Figure 9b. Nevertheless, using this approach of ion channel modeling, the ion channel could be simulated as a single system or a system in a control loop with any randomly chosen input, enabling the obtained output to be analyzed.

**Figure 9.** Simulink Kv10.1 model in a control loop (**a**); simulated output for sine input (**b**).

Microelectromechanical (MEMS) devices and nanoelectromechanical (NEMS) devices can be designed as MEMS PID controllers and NEMS PID controllers. In a clinical context, nanorobots are already used for the mechanical opening of a cell membrane [32], microscopic surgical procedures, drug delivery, gene therapy, or selective destruction of cancer cells [4]. Therefore, our vision is that free-floating and guided nanorobots could have promising applications in cancer detection and therapy. Research has shown that the up-regulation of the selected channel Kv10.1 can promote cell proliferation and thus tumor development in different organs and human body systems [18], whereas the downregulation of Kv10.1 could potentially have the opposite effect, providing a potential oncological target. However, since the same ion channels are also expressed in normal, healthy cells, albeit to minor expression levels, one of the biggest challenges is to specifically block or manipulate them only in cancer cells. Nanorobots could be programmed to target certain cell clusters, tissues or organs (e.g., prostate, stomach, or breast) and specifically control the activity of selected ion channels in cancer cells without affecting the same channels expressed in surrounding healthy cells or other tissues and thus impairing their function. A controlled down-regulation of the ion channel current in specific cells only would also allow for a systemic application, such as the use of traveling nanorobots for the treatment of cancer in various regions of the body or targeted tissues and cell clusters, as shown in Figure 10, for example.

#### *4.3. Potential Use Case for Treating Breast Cancer Using a Kv10.1 Controlled Nanorobot*

Our hypothesis for the use of swimming nanorobots is to control ion-channel activity to specifically target cancer cells in breast tumors. We know that Kv10.1 channels in the nervous system contribute to the control of neuronal excitability. However, since activation requires strong depolarization, a single action potential would not be sufficient for activation [33]. In contrast, cancer cells are usually depolarized so that the channels are predominantly open, providing a potential target for pinpoint destruction by nanorobots. For example, overexpression of Kv10.1 channels is significantly up-regulated in invasive breast carcinoma, whereas healthy breast tissue shows hardly any expression or activity of the Kv10.1 channel [30,33–36]. Floating nanorobots applied via the blood circulation can be guided to the targeted sites and measure and control the Kv10.1 channel activity of the cells in the area of interest. Since nanorobots act as sensors and actuators, if the sensor interface detects an altered behavior of Kv10.1 channel activity, as is the case in breast cancer cells, the channels could be controlled accordingly by the Kv10.1 feedback control system of the robot (see Figures 9 and 10). Alternatively, after detecting a cancer cell based on aberrant or

abnormal channel activity, such MEMS/NEMS-based nanorobot devices may also act as a nano-scalpel that mechanically destroys the targeted cell.

**Figure 10.** Illustration of possible application of Kv10.1-controlled nanorobots in breast cancer detection and treatment. The cancerous area will be reached by the nanorobots through the blood vessel system. Created with BioRender.

Further examples of possible applications are discussed in [34,37–39]. However, although the practical implementation of this use case has not been performed and validated, this work demonstrates a potential new, forward-looking direction for cancer detection and therapy.

#### **5. Conclusions**

In this work, voltage-gated ion channels were modeled as control objects or systems by applying the control system theory and a system identification method. The Kv10.1 voltage-gated ion channel exhibits dynamic behavior as a first-order dynamic system. We presented a classical control algorithm and described how this algorithm could theoretically be used to control the response of the Kv10.1 voltage-gated ion channel. Other control algorithms, such as robust control or fuzzy control, could also be used. Research has shown that controlling the Kv10.1 ion channel under up-regulated conditions plays a vital role in cancer prevention and treatment. Nanoelectromechanical systems, such as nanorobots, could be used in the treatment of cancer as they are able to travel through the blood vessel system to the target cells and bind to the ion channels. The robot's control unit can then be activated based on the implemented control algorithm, and the desired ion channel regulation can be performed to treat channel activity by the down or upregulation of the targeted channels.

It is, therefore, expected that such models, which could also describe the activity of multiple ion channels and their interactions, e.g., described by a MIMO system, will enable the use of nanoelectromechanical systems and nanorobots, as demonstrated for the Kv10.1 voltage-gated ion channel, and for other ion channel families, with the goal of establishing new tools for cancer detection and treatment at the single-cell level.

**Author Contributions:** Conceptualization, J.L.Š.; methodology, J.L.Š.; validation, J.L.Š.; formal analysis, J.L.Š.; investigation, J.L.Š. and S.L.; resources, C.B.; data curation, J.L.Š. and S.L.; writing—original draft preparation, J.L.Š.; writing—review and editing, J.L.Š., S.L. and C.B.; visualization, J.L.Š.; supervision, J.L.Š. and C.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** Open Access Funding by the Graz University of Technology.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** https://channelpedia.epfl.ch/ionchannels/ (accessed on 15 February 2022).

**Acknowledgments:** We acknowledge Open Access Funding by Graz University of Technology.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

