Electrothermal Effects on Biomolecular Binding Kinetics: A Medical Modeling Approach with PID Control

1. Introduction

During the last century, several natural disasters generated an efficient strategy and management to prepare and prevent any emergency [1, 2]. In fact, the most dangerous situation is related mostly to the pandemic disease which stretches to almost every element of society [3]. However, the public health system is the first wall against any pandemic situation which is responsible for understanding and analyzing its spread and impact on human health as can be seen with the Spanish flu (H1N1, 1918), Asian flu (H2N2, 1957), Ebola Virus (2014) pandemic flu (H1N1, 2009) [4]. Among the different aspects and virus genome, Coronavirus is one of the most dangerous infections related to severe respiratory diseases with two types of virus known as Severe Acute Respiratory Syndrome Coronavirus (SARS-CoV) emerged in 2002 in Guangdong Province, China which caused 774 deaths and 8098 infected cases, meanwhile, the second type known as Middle East Respiratory Syndrome Coronavirus (MERS-CoV) appeared in Saudi Arabia in 2012 and touch 27 countries with 35 % of mortality rate [5].

Recently, a world pandemic disease identified as a novel coronavirus related to severe acute respiratory syndrome coronavirus (SARS-CoV-2) was an outbreak in particular from Wuhan, Hubei Province, China in December 2019 and then spread around the world associated with 6.520.263 deaths in September 2022 [6]. The World Health Organization (WHO) defines this biological disaster as coronavirus disease 2019 (COVID-19) which can be linked to the family of Coronaviridae and subfamily Orthocoronavirinae belongs to the same Betacoronavirus genus with 80% and 50% similarity to SARS-CoV and MERS-CoV, respectively [7]. In fact, this new virus has a spherical structure which presents the biggest size around 120 nm and its single-stranded RNA genome is characterized by positive sense and size ranges from 26 to 32 kb [8]. This later is packaged into a capsid which is constituted by the nucleocapsid protein N and enveloped by a membrane that contains three structural proteins known as the membrane protein (M) and the envelope protein (E) and the spike glycoprotein (S) [9]. In fact, the spike (S) proteins present the key to making a specific binding with the cellular entry receptors to penetrate into the host cell [10]. However, the intracellular life cycle of the infection starts with the destruction of the target cell and then leaves to attack its neighbors. In fact, SARS-CoV-2 has attracted the intension of the social community because of the high spread and dangerous symptoms from mild to severe such as fatigue, cough, fever, and respiratory problems [11]. Moreover, mental health and psychological disorders including depression, anxiety, and stress persist during the treatment and quarantine [12]. It should be mentioned also that the high spread and high mortality rate of the pandemic yield to a devastating impact on all aspects of economic performance and business activities [13].

Early detection of the covid-19 using new devices of biosensors based on nanotechnologies systems becomes an important issue that can be used to stop or decrease the speed of the pandemic’s spread. Usually, the performance of a wide range of biological detection can be controlled and examined by affinity, specificity, and sensitivity which are the most important point to design biosensors with high quality [14]. In fact, researchers exploited the congruence of the phylogenetic phase characterized by the replication of the SARS-CoV-2 in the lung parenchyma of the covid-19 and SARS as well as MERS to design adequate biosensors. Respective diagnostic methods, types of biological systems (throat swab/plasma/nasal), and fixing specific biomarkers are important parameters associated with the particular virus [15]. Usually, the genome and protein of the infection are the most biomarkers for detection of SARS-CoV-2.

As mentioned by the world health organization (WHO), the diagnostic process applied in Covid-19 detection is already associated with the quantitative real-time polymerase chain reaction (qRT-PCR) [16]. Despite the frequent errors, highly expensive and long-lasting analysis, this method can be used also for a small number of infected cells due to the RNA amplification of SARS-CoV-2. Polymerase chain-reaction (PCR) presents the standard technique used to design biosensors for Covid-19 but its false result in some cases yields to use of a complementary diagnostic assay [17]. However, serological tests are a technique of SARS-CoV-2 detection applied usually after two weeks of the onset of the symptoms which is related to the IgG/IgM antibodies and meta-analysis counting enzyme-linked immunosorbent assay (ELISA), fluorescence immunoassays (FIA) and lateral low immunoassays (LFIA) [18]. However, these techniques cannot be in the early stage of disease and required training in relatively complex laboratories. Recently, to overcome the disadvantage of (qRT-PCR) and serological, several design biosensors with different mechanisms based on optical, electrochemical, and colorimetric modalities have been proposed as an alternative system [19]. However, emerging point-of-care (POC) diagnostic, reverse transcription loop-mediated isothermal amplification (RT-LAMP), nano-pore targeted sequencing (NTS) method presents an important technology for immunoassays application and can be the next generation of the SARS-CoV-2 biosensors with rapid detection, portable form, low cost, and simple moderation [20].

Biosensors have a crucial role in healthcare strategy and their performance is the key to avoiding a spread disaster. In fact, the performance of biosensors is proportional to the surface binding reaction evaluated by the long detection and time diffusion rate [21]. Therefore, understanding and application of some physical parameters such as hydrodynamic pressure, magnetic effect, optical forces, and AC electro-kinetics pimp which include many processes such as AC electro-osmosis (ACEO), and AC electro-thermal effects (ACET) are important to improve the microfluidic agitation and kinetics reaction which are the main purpose of many reports to raise the binding reaction rate of the biosensors [22-24]. In biomedical devices, Qinlong Ren et.al [25] report that the electro-thermal effect (ACET) is pronounced by the variation of dielectric constant and conductivity with temperature related to the Joule heating effects or other sources which create a direct impact on micro-flows and particle motions. This improves the kinetic of physiological fluids and the binding between the hydrophilicity of the sensor surface and the analyte [26]. However, ACET performance and the rate of binding reaction in heterogeneous immunoassays can be enhanced by the frequencies, electrical conductivity, and applied voltage. The present work is devoted to introducing a valuable and computational model that was applied to resolve the coupled set equations involving Navier- Stokes equations with convection, diffusion, and reaction of targeted molecules based on FEM. This model can simulate the effect of the ACET technique on the SARS-CoV-2 S protein binding reaction without the need to modify the input values of the applied voltage by using PID control. Especially, the temperature rise due to the applied ACET on the binding reaction was controlled and the applied voltage was deduced using PID. Additionally, several important parameters were evaluated and investigated, including the effect of electrothermal, several inlet analyte concentrations, and inlet velocities on the binding reaction.

2. Materials and methods

A robust, sensitive, reliable, efficient, low-cost, and quick-response testing biosensor is the most interesting solution for the reduction of death in pandemics [27]. Moreover, the improvement of their performances still remains a challenge. Nevertheless, several numerical and experimental studies are still constantly evolving [28]. From the side of the biosensor, the most significant part of the design is associated with the transport of the analytes to the sensitive surface of the biosensor due to the developed diffusion boundary layer [29]. The computation of the impact of the applied electro-thermal field on the binding reaction is one of the best ways to enhance the biosensor [30]. Fig. 1 exhibits a schematic representation of the biosensor principle with the use of computational fluid dynamics.

Figure

Fig.1. Schematic representation of the biosensor principle with the use of computational fluid dynamics.

Geometrical and physical parameters: The geometry considered in this work consists of a rectangular micro-channel 500 µm × 150 µm) characterized by a functionalized membrane 40 µm × 3 µm (length× width) located on the top wall. In fact, two electrodes fixed on the bottom induce an electric field that generates an electro-thermal force.

Temperature filed -controlled modeling:

In medical applications, the temperature rise should be controlled precisely when using the ACET for the immunoassay reaction. Therefore, the ACET controlled-temperature rise eliminates any negative effects and provides the kinetic reaction without considering the variation of electric field and fluid properties with temperature [30].

In this study, a programmable temperature-controlled based on the proportional-integral-derivative (PID) controller was employed to control and manipulate the variable voltage source. This application moderates the variable voltage source to reach the maximum allowable temperature rise which can be around 5 K for biochemical system under normal physiological conditions. The input voltage has been calculated by:

(1)

where is the applied input voltage, , and are proportional, integral, and derivative coefficients respectively [31].

Microfluidic immunoassay reaction

The estimation of the analyte concentration [A] is required to compute the complex Antibody-S protein concentration [AB] [32]. In fact, these quantities control the analyte transport process from the bulk to the fixed reaction surface by fluid flow velocity which depends on the temperature and the electro-thermal force. The second process that should be present in the biosensor mechanism is related to binding reaction and the chemical interaction between A and B ligand [33]. Therefore, the overall model couples between the convection and diffusion application mode with the reaction surface boundary condition.

Potential and the electrical field :

Quasi-static regime approximation and Poisson equation were numerically applied to get the potential and the electrical field values. In fact, the electrical neutrality of the fluid and the permittivity dependency with temperature can be neglected in the Poisson equation which can be expressed as bellow [34]:

and (2)

However, the boundary conditions that should be taken into consideration for the electrostatic system are related to electrode positions. In fact, a variable input electric potential is applied near the electrodes (left and right) computed by PID controller, meanwhile the electric insulation condition is assumed for the other boundaries.

Temperature and the velocity fields: We suppose that the variation of the thermal conductivity and specific thermal capacity against temperature can be omitted in the follows energy equation [35, 36]:

(3)

Moreover, due to the small value of the viscous dissipation compared to the Joule effect by factor of 10-8, we have neglected the viscous dissipation of the fluid in the micro-channel [30]. This later was assumed to be incompressible with a Newtonian feature characterized also by a steady and laminar flow in the micro-channel but is not isotherm. Using Navier-Stokes equations, we are able to calculate the flow velocity field. The continuity and motion of the fluid can be described by the equations:

(4)

(5)

where p denotes the pressure, Fe designs the electro-thermal force and represents the dynamic viscosity of the fluid in the 2D-system. The physical parameters and intrinsic behavior of the micro-channel fluid such as the fluid density (ρ=1000 kg/m3) and the kinematic viscosity (μ=10-6 m2/s) are considered to be constant.

The two electrodes induce a variation of temperature arises from the joule heating in the micro-channel. The applied temperature gradient causes a local modification in the electrical permittivity ε and conductivity σ of the medium which generate a motion of the micro-fluidic system by the applied electro-thermal force defined by the general expression [36, 37]:

(6)

where τ andrepresents the charge relaxation time and the angular frequency of the applied electrical field, respectively. Moreover, the electro-thermal force can be estimated for aqueous media around 293 K by considering the framework of Green et al. [35] as below reported:

Thus, the electro-thermal force is expressed by:

(7)

In fact, the applied heating flux and electro-thermal force condition are used to describe the outlet/inlet system of the micro-channel whereas the rest elements of the walls are supposed thermally insulated. The inlet micro-fluid flows longitudinally on the x-axis with a parabolic velocity , where and H denotes the average inlet flow velocity and the micro-channel height, respectively. The fluid is supposed to be at rest initially and the channel outlet is open to the atmosphere while other solid boundaries are admitted to be non-slip boundaries.

The inlet section is characterized by a temperature identical to the ambient temperature T0 and we assumed also that the outlet part is in the heat flux condition. Moreover, the electrode sample is supposed to be a perfect heat conductor so that its temperature is identical to the ambient temperature T0 for the designed biosensors.

The binding reaction: the biosensor detection is performed also by the binding reaction process and the diffusion flux of the analyte fraction to the sensitive membrane which assures the interaction with the localized antibody ligand B. In fact, at the reaction surface, the binding reaction takes place and this produces the analyte-ligand complex AB:

where, [B], and [AB] are the analyte, the antibody ligand, and the complex concentrations at the sensitive membrane. The analyte portion is the unique parameter that should be diffused in the channel of the biosensors while the antibody ligand B and the complex AB are fixed on the reaction surface.

To analyze the interaction of the immobilized antibody with the analyte, the first-order Langmuir adsorption model was used to explain the binding reaction. The reaction rate is simulated by the chemical kinetics expression:

(8)

Where and denote the association and dissociation rate constants, respectively. The concentration of free antibodies is denoted by .

Convection and diffusion of the analyte: The transport of the analyte molecules SARS-COV-2 by convection and diffusion towards the membrane is described by this equation

(9)

Where D denotes the coefficient of the analyte diffusion, represents the flow velocity and G is the reaction rate. This latter is negligible due to the absence of the reaction at the micro-fluid bulk and we assumed that it is equals to zero in this work.

To estimate the analyte concentration, several boundary conditions and approximations are used to solve the convection-diffusion equation.

The left boundary is fixed to a constant concentration . At the right boundary, we used the condition , it is mean that the remaining concentration leaves the system by convection, and at the reaction surface, a diffusive flux is imposed. It is given by:

The other parts of the walls are impermeable without any interaction with the analyte. The initial condition of the biosensor is performed by the absence of the analyte concentration at the reaction surface. For the binding reaction (i.e. Eq. 9). The initial condition is fixed .

Numerical method mesh sensitivity analysis

The numerical model uses four application modes: electrostatics, convection and conduction, incompressible Navier-Stokes and convection and diffusion to simulate the buffer fluid flow and uses the diffusion mode to simulate the binding reaction of the analyte with the ligand immobilized on the reaction surface of the biosensor

For the present system, a suitable numerical approach based on the framework of Galerk finite element method (FEM) was utilized to solve the coupled governing equations such as the transport mechanism combined with the first Langmuir adsorption model. Analyses of the physical mechanism coupled with the proposed model and equation was applied to retrieve a numerical solution using a computer developed code. For the geometric design of the biosensor, we assume that the two-dimensional simulation domain is subdivided into triangular elements. The refinement of the electrodes and the reaction surface regions was manipulated with a high mesh quality and all variables are approximated by a polynomial in each element.

3. Results and discussion

During chemical reaction, the input voltage parameters into the biosensor should be controlled to keep the authorized condition and to avoid any damage of the biofluid nature. In fact, a voltage-controlled program combining the proportional-integral-derivative (PID) controller algorithm was developed in this simulation to keep the input voltage below a limit value [38]. The optimal voltage (Vmax = 10.29 V) corresponds to the steady-state output of the PID controller required to maintain the maximum allowable temperature rise (ΔT < 5 K). This value was determined through iterative numerical simulations in which the controller continuously adjusted the applied voltage to satisfy the prescribed thermal constraint.

3.1. Effect of the applied electric field on the binding reaction

In this part, we analyzed the influence of the electric field on the kinetics of the binding reaction. In fact, the emitted electric field through the two electrodes warms up the fluid into the micro-channel due to the electro-thermal effect and Joule heating [39]. This situation creates heterogeneous heating of the fluid and a local modification in conductivity and permittivity which are the major reason of the electro-thermal force. However, the electro-thermal force produces a vortex motion of the fluid which increases the flow stir and reduces the diffusion boundary layer thickness. In fact, the electro-thermal force can help the analyte to reach the sensing surface in a much shorter time which considerably enhances the concentration and kinetics of the binding reaction. Thus, the electro-thermal effect is an extremely useful tool for biological fluid manipulation [40].

Fig.2 presents the simulated data of the surface complex concentration versus time for a micro-channel biosensor. Obviously, the detection time dramatically fall with 63% percentage under the effect of AC-electric field and conditional applied voltage. This behavior is attributed to the important kinetics rate of the binding reaction SARS-CoV-2 S protein and the response time of heterogeneous immunosensors Table 1.

Figure

Fig.2 Surface complex concentration in the cases without and with applied electric field Table. 1. Detection time, temperature rise and drop percentage in case without and with applied electric field.

Used voltage (V)	Detection time (s)	Temperature rise (K)	Drop percentage (%)
0	3480	/	/
10.29	1279	5	63

Fig.3 shows the distribution of the analyte concentration in the absence (a) and presence (b) the electric field application. It can be seen that the distribution of the analyte concentration in the microfluidic channel depend on the presence of the electric field. In fact, the electro-thermal force generated by the applied electric field destroy the uniform distribution behavior of the analytes concentration in the micro-channels Fig. 3 (a) and improves the mass-transport of the analytes in the vicinity of the sensitive area of biosensor as shown in Fig. 3 (b).

Figure

Fig.3 Analyte concentration distribution (a) in the absence of an electric filed and (b) in the applied an electric field.

3.2. Effect of the inlet analyte concentration

The left analyte concentration is an important factor used to overcome the deficient of consumed analyte near the reaction surface [41]. Fig.4 shows the effect of various SARS-CoV-2 concentrations on the binding reaction without and with the applied electric field. This figure confirms that the association and dissociation phases have different characteristic behaviors. Also, we can see that the increase of the analyte concentration at the inlet from 1 μmol/m3 to 9 μmol/m3 yield a significant reduction of the response time from 4659 s to 2465 s, respectively. Moreover, we can see from the Fig.4 that the application of electric field enhances the binding reaction and further decreases the detection time of the biosensor with 63%. All the numerical results listed in Table 2 confirms that the electro-thermal effect and time detection is proportional to the inlet analyte concentration and applied electric field.

Figure

Fig.4. Effect of various SARS-CoV-2 concentrations the binding reaction in case without and with applied electric field.

Table. 2. Detection time and drop percentage for various SARS-CoV-2 concentrations in case without and with applied electric field.

Used SARS-CoV-2 concentration (µmol/m3)	Detection time (s)in the case with electrothermal effect	Detection time (s)in the case without electrothermal effect	Drop percentage (%)
1	4050	4659	13
3	2549	4384	41
6	1333	3568	62
9	899	2465	63

3.3. Effect of the inlet flow velocity

The inlet flow velocity is an important factor in the binding reaction which can reduce the effect of the thickness of the diffusion boundary layer into the micro-channel of biosensor. Indeed, Fig. 5 denotes the temporal variation of surface complex concentration versus time at various velocities without (a) and with (b) applied electric field. As can be seen from Fig.⁵ (b), we found that the time detection declines with the increase of velocity in absence of the electric field while Fig.⁵ (a) show almost an independent behavior with the velocity variation under the electric field. In fact, raising the flow velocity increase the complex concentration and SARS-CoV-2 S protein/antibody binding reaction as already mentioned and the presence of the electric field yields to a further decrease of the response time as listed in Table 3.

Figure

Fig.5. Effect of various velocities on the SARS-Con-2 binding reaction in case without and with applied electric field.

Table. 3. Detection time and drop percentage for various velocities in cases without and with applied electric field.

Velocity (µm/s)	Detection time (s)in case with electrothermal effect	Detection time (s)in case without electrothermal effect	Drop percentage (%)
100	1333	3568	62
200	1323	3022	56
400	1309	2465	46
800	1301	2039	36

Different configuration and parameters were applied to enhance the effect of the biosensors and our result using the inlet analyte concentration, inlet flow velocity show that application of the electric field can highly decrease the response time as revealed in Figs 4 and 5. Thereby, the application of the electric field is an important method to enhance the binding reaction.

4. Conclusion:

Early detection of the covid-19 represents a major challenge for new devices of biosensors based on nanotechnology systems which becomes an important issue that can be used to decrease the speed of the pandemic spread. Our numerical result proves that the performance of biosensors is related to the effect of applied alternating current electro-thermal (ACET) force on the immunoassay reaction during temperature-controlled. In fact, the heating and temperature with their crucial roles were controlled by a proportional-integral-derivative (PID) controller to avoid any damage or false detection. In addition, it should note that temperature of the microfluidic biosensors can moderate the biosensors response. This system proves that the applied voltage should not exceed its optimal value of Vmax=10.29 V corresponding to the limit value of heating (T < 5 K). Different biosensor factors such as electro-thermal force, inlet flow velocity, electric field and analyte concentration were analysed and prove their influence to enhance the binding reaction and reduce the time detection of the biosensors.

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