BioFET-SIM: A Tool for the Analysis and Prediction of Signal Changes in Nanowire-Based Field Effect Transistor Biosensors

Abstract

Biosensors based on nanowire field effect transistor (FET) have received much attention in recent years as a way to achieve ultra-sensitive and label-free sensing of molecules of biological interest. The BioFET-SIM computer model permits the analysis and interpretation of experimental sensor signals through its web-based interface www.biofetsim.org. The model also allows for predictions of the effects of changes in the experimental setup on the sensor signal. After an introduction to nanowire-based FET biosensors, this chapter reviews the theoretical basis of BioFET-SIM models describing both single and multiple charges on the analyte. Afterwards the usage of the interface and its relative command line version is briefly shown. Finally, possible applications of the BioFET-SIM model are presented. Among the possible uses of the interface, the effects on the predicted signal of pH, buffer ionic strength, analyte concentration, and analyte relative orientation on nanowire surface are illustrated. Wherever possible, a comparison to experimental data available in literature is given, displaying the potential of BioFET-SIM for interpreting experimental results.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

3.1 List of Abbreviations and Symbols

FET:

Field Effect Transistor

r _nw :

Nanowire radius

r _ox :

Oxide layer thickness

r _bf :

Biofunctionalization layer thickness

ε :

Free space permittivity

ε ₁ :

Nanowire permittivity

ε ₂ :

Oxide permittivity

ε ₃ :

Buffer permittivity

μ :

Nanowire charge carrier mobility

n ₀∕p ₀ :

Nanowire (electron/hole) charge carrier density

λ _TF :

Nanowire Thomas–Fermi screening length

λ _D :

Buffer Debye screening length

e :

Unit of charge

ℏ :

Plank’s constant divided by 2π

m ^∗ :

Charge carrier effective mass

k _B :

Boltzmann’s constant

N _A :

Avogadro’s number

3.2 Introduction

Nanowire-based field effect transistor (FET) biosensors have recently been the subject of much research and development. A sensor is a device that permits the recognition of an analyte in a sample through some form of interaction and the generation of a signal that can be recorded. According to [1] a biosensor is “a self-contained integrated device, which is capable of providing specific quantitative or semi-quantitative analytical information using a biological recognition element (biochemical receptor) which is retained in direct spatial contact with an electrochemical transduction element.” Nanoscale materials offer a large surface-to-bulk ratio and thus a high sensitivity towards changes in the charge distribution near the surface and are thus highly interesting for sensing. Nanobiosensors are characterized by a transduction element of nanoscale dimensions, like nanotubes, nanoribbons, or nanowires. The latter is the subject of this chapter.

Nanowire-based biosensors work as FETs: the current flowing through the semiconducting nanowire material is perturbed by the sensing event involving a charge redistribution at the surface of the nanowire. Generally with an n-type nanowire a positive surface charge produces a positive signal (i.e., higher conductance) and vice versa. The opposite is true for a p-type nanowire. In an n-type (p-type) nanowire the charge carriers are electrons (holes). The change in surface potential created by a charge attracts or repels the charge carriers, creating the change in conductance. In a nanowire-based biosensor (for simplicity only called biofet from now on) a nanowire resides on an insulating substrate and is electrically contacted at its extremes, Fig. 3.1. Usually a second insulating layer, commonly constituted by an oxide, is deposited on top. Finally the analyte recognition layer is created through chemical modifications of the surface. The latter layer contains the recognition element and is responsible for specific capturing of the desired analyte and is usually referred to as a biofunctionalization layer. Examples of a biofunctionalization layer are a biotinylated surface for sensing of avidin or streptavidin [2, 3], or an antibody modified surface for the recognition of a given antigen [4].

Biofets have been employed or studied for pH [5], protein [3, 6–8], and DNA sensing [9, 10], for blood analysis [11] and for applications in nanotechnology-based medicine [12]. See [13–17] for a more thorough review of biofets and [18–20] for studies on their performance limits.

Fig. 3.1

Generic representation of a biofet. An insulating substrate carries a nanowire (transduction element) contacted at its extremes. An oxide layer is deposited on top. Biochemical receptors are chemically bond to form the recognition element capable of specifically recognizing the desired analyte

In addition to the experimental work, a series of numerical theoretical models have been presented to evaluate the performance of biofets [21–27]. Based on this theoretical work, tools for simulating biofets have been made available [28–30]. Following the work of Sørensen et al. [31] we developed our own model for simulation of the conductance sensitivity of a biofet in the presence of an analyte represented by a single charge [32]. We named our model BioFET-SIM. Section 3.2 gives a description of the theoretical background of BioFET-SIM and the approximations involved. BioFET-SIM has been subsequently extended to simulate the signal given by multiple charges approaching the surface of a biofet [33]. The usage of a multiple charges model proved essential when simulating the capture of analytes with an asymmetric charge distribution. Recently BioFET-SIM has been further extended [34] to include explicitly the biofunctionalization layer and made available through a web interface hosted at www.biofetsim.org. The main advantage of the web interface is the possibility to graphically interact with the sensed analyte, in order to define the specific orientation of it with respect to the nanowire surface. Further details on the usage of BioFET-SIM are given in Sect. 3.3 and some possible applications are shown in Sect. 3.4. Thanks to its simplicity of use BioFET-SIM is a tool successfully employed in present day research [4, 35–42].

Fig. 3.2

Graphical representation of a generic nano BioFET. A nanowire of length L and radius r _nw is coated with an oxide layer of thickness r _ox. On the surface a biofunctionalization layer of thickness r _bf is responsible for capturing the analyte, represented by the charges q. The entire system is immersed in a buffer solution

3.3 Theoretical Background

In this section we will review equations and assumptions behind the BioFET-SIM model.

3.3.1 Base Conductance

The BioFET-SIM model describes nanowire-based biosensors with elements as depicted in Fig. 3.2. The BioFET-SIM model predicts the conductance sensitivity Δ G∕G ₀ where Δ G is the difference between the conductance upon binding, G, and the base conductance, G ₀, of the nanowire. The bulk conductance G ₀ of a homogeneous cylindrical nanowire can be expressed as:

$$\displaystyle{ G_{0} = \frac{\pi \,r_{\mathrm{nw}}^{2}\,e\,(n_{0}\,\mu _{n} + p_{0}\,\mu _{p})} {L} }$$

(3.1)

where r _nw is the radius of the wire, e the elementary charge, n ₀ the electron density, μ _n the electron mobility, p ₀ the hole density, μ _p the hole mobility, and L the length of the nanowire. For a nanowire material with high enough doping, (3.1) can be approximated by considering only the electron or the hole density (only n ₀ or p ₀) and mobility (indicated simply as μ). For example, for a doped p-type nanowire, it is possible to rewrite (3.1) as:

$$\displaystyle{ G_{0} = \frac{\pi \,r_{\mathrm{nw}}^{2}\,e\,p_{0}\,\mu } {L} }$$

(3.2)

In our model only one type of charge carrier for the entire nanowire is considered, and the nanowire is assumed to resemble a low density material. Δ G∕G ₀ is evaluated through a Thomas–Fermi screening model of the charge carriers [43–45].

A major assumption governs the description of the charge carrier concentration, that is the charge carrier concentration is distributed uniformly through the nanowire. This implies that effects of, e.g., charges trapped in the oxide layer are not considered as influencing the charge carrier distribution. One has to remember that, formally, n ₀ (p ₀) represents the initial charge carrier density.

Moreover, we assume here that effects related to the metal contacts are negligible.

3.3.2 Conductance Sensitivity

Following [31] one can define the generic sensitivity of a p-type doped nanowire as:

$$\displaystyle{ \frac{\Delta G} {G_{0}} = -\, \frac{2} {r_{\mathrm{nw}}\,e\,p_{0}}\varGamma \,\sigma _{\mathrm{s}} }$$

(3.3)

where σ _s is the “sensed” charge density at the surface of the nanowire and Γ is a dimensionless function quantifying the actual sensitivity of the nanowire. In case of an n-type nanowire (3.3) becomes:

$$\displaystyle{ \frac{\Delta G} {G_{0}} = \frac{2} {r_{\mathrm{nw}}\,e\,n_{0}}\varGamma \,\sigma _{\mathrm{s}} }$$

(3.4)

In the special case of a cylindrical nanowire it is possible to express Γ analytically as [46]:

$$\displaystyle\begin{array}{rcl} \varGamma = \frac{\epsilon _{1}\,K_{0}\left (\frac{r_{\mathrm{nw}}+r_{\mathrm{ox}}} {\lambda _{\mathrm{D}}} \right )\, \frac{\lambda _{\mathrm{D}}} {\lambda _{\mathrm{TF}}} \,I_{1}\left (\frac{r_{\mathrm{nw}}} {\lambda _{\mathrm{TF}}} \right )} {\left [K_{0}\left (\frac{r_{\mathrm{nw}}+r_{\mathrm{ox}}} {\lambda _{\mathrm{D}}} \right )\,\left ( \frac{\lambda _{\mathrm{D}}} {r_{\mathrm{nw}}+r_{\mathrm{ox}}} \right ) +\ln \left (\frac{r_{\mathrm{nw}}+r_{\mathrm{ox}}} {r_{\mathrm{nw}}} \right )\,K_{1}\left (\frac{r_{\mathrm{nw}}+r_{\mathrm{ox}}} {\lambda _{\mathrm{D}}} \right )\,\frac{\epsilon _{3}} {\epsilon _{2}} \right ]\,\epsilon _{1}\,\left (\frac{r_{\mathrm{nw}}} {\lambda _{\mathrm{TF}}} \right )\,I_{1}\left (\frac{r_{\mathrm{nw}}} {\lambda _{\mathrm{TF}}} \right ) +\epsilon _{3}\,K_{1}\left (\frac{r_{\mathrm{nw}}+r_{\mathrm{ox}}} {\lambda _{\mathrm{D}}} \right )\,I_{0}\left (\frac{r_{\mathrm{nw}}} {\lambda _{\mathrm{TF}}} \right )}& &{}\end{array}$$

(3.5)

where λ _D is the Debye screening length of the buffer, λ _TF the Thomas–Fermi screening length typical of the nanowire material, ε ₁, ε ₂, and ε ₃ are the relative permittivities of the nanowire, the oxide layer, and the buffer, respectively, r _ox is the oxide layer thickness, I ₀, I ₁, K ₀ and K ₁ are the modified Bessel functions of first and second kind, respectively [47]. Γ is a dimensionless function with values ranging from zero to unity.

3.3.3 Single Charge Model

The presence of additional charges at a distance l from the nanowire surface gives rise to a charge density σ _b. Equation (3.3) is rewritten as:

$$\displaystyle{ \frac{\Delta G} {G_{0}} = -\, \frac{2} {r_{\mathrm{nw}}\,e\,p_{0}}\varGamma \left (\varGamma _{l}\,\sigma _{\mathrm{b}} +\sigma _{\mathrm{s}}\right ) }$$

(3.6)

The BioFET-SIM model concerns the simulation of a signal in the conductance when an analyte is attached to the surface. The analyte is represented by σ _b and it is supposed not to modify the charge density σ _s already present at the nanowire surface. The conductance given by σ _s is considered as part of the background signal. For this reason the σ _s term can be dropped from (3.6) that can be written as:

$$\displaystyle{ \frac{\Delta G} {G_{0}} = -\, \frac{2} {r_{\mathrm{nw}}\,e\,p_{0}}\varGamma \left (\varGamma _{l}\,\sigma _{\mathrm{b}}\right ) }$$

(3.7)

and Γ _l is defined as:

$$\displaystyle{ \varGamma _{l} = 2 \frac{r_{\mathrm{nw}}} {r_{\mathrm{nw}} + l}{\left (1 + \sqrt{ \frac{r_{\mathrm{nw } } } {r_{\mathrm{nw}} + l}}\exp (l/\lambda _{\mathrm{D}})\right )}^{-1} }$$

(3.8)

where l is the distance of the sensed charge from the nanowire surface, inclusive of the biofunctionalization layer (r _bf). If the analyte is approximated as one charge as in Fig. 3.3a, (3.5), (3.7), and (3.8) describe the single charge BioFET-SIM model [32].

Fig. 3.3

PROPKA computed charges at pH = 7.4 for streptavidin (see Sect. 3.2.8). (a) The overall charge is assumed to reside at the center of mass of the protein (single charge model). (b) Each ionizable amino acid carries its charge (multiple charges model). Blue, white, and red circles correspond to a positive, neutral, and negative charge, respectively

3.3.4 Multiple Charges Model

A complex analyte system like a protein can give rise to unexpected signals because of its nonuniform charge distribution [48]. In these cases a single charge description of the sensing event is not sufficient. Considering multiple charges, the sensitivity is evaluated as:

$$\displaystyle{ \frac{\Delta G} {G_{0}} = - \frac{2} {r_{\mathrm{nw}}ep_{0}}\varGamma \left [\sum _{i}^{m}\left (\varGamma _{ l_{i,\mathrm{tot}}}\sigma _{b_{i}}\right )\right ] }$$

(3.9)

and $\varGamma _{l_{i,\mathrm{tot}}}$ is defined as:

$$\displaystyle{ \varGamma _{l_{i,\mathrm{tot}}}\left (\lambda _{\mathrm{D}}\right ) = 2 \frac{r_{\mathrm{nw}}} {r_{\mathrm{nw}} + l_{i,\mathrm{tot}}}{\left [1 + \sqrt{ \frac{r_{\mathrm{nw } } } {r_{\mathrm{nw}} + l_{i,\mathrm{tot}}}}{e}^{l_{i,\mathrm{tot}}/\lambda _{\mathrm{D}} }\right ]}^{-1} }$$

(3.10)

where l _i, tot is the overall distance of the charge q _i from the surface of the nanowire. Considering Fig. 3.2, the distance l _1, tot relative to q ₁ is computed as:

$$\displaystyle{ l_{1,\mathrm{tot}} = r_{\mathrm{ox}} + r_{\mathrm{bf}} + l_{1} }$$

(3.11)

Equations (3.5), (3.9), and (3.10) constitute the multiple charges BioFET-SIM model [33]. Figure 3.3b reports the multiple charges description of streptavidin. Each blue, white, and red circles represents a positive, neutral, and negative q _i charge, respectively. A more detailed description of how BioFET-SIM computes l _i, tot is given in [34].

Note that we approximate the charge distributions of the charges as homogeneous over the nanowire surface, and only varying outwards perpendicular to the nanowire surface. This approximation enables analytical solutions, very fast calculations, and high-throughput simulations.

3.3.5 Thomas–Fermi Screening Length

In the BioFET-SIM model we settle on a rough description of the electronic structure in the nanowire reflecting our lack of detailed knowledge about the complex situation at its surface. The possible deactivation of dopants at the surface [49] or the increased dopant concentration near the surface compared to the semiconductor bulk [50] is approximated by the sole Thomas–Fermi screening length λ _TF [32]. For the p-type (n-type) nanowire, the screening length λ _TF is related to the charge carrier density p ₀ (n ₀) through:

$$\displaystyle{ \lambda _{\mathrm{TF}} = \sqrt{ \frac{{\hslash }^{2 } \epsilon {_{r } \pi }^{4/3 } } {{m}^{{\ast}}{e}^{2}p_{0}^{1/3}}} }$$

(3.12)

where ε _r is the relative permittivity of the nanowire material (ε ₀ ε ₁) and m ^∗ is the effective mass of the charge carrier in units of an electron mass at rest. p ₀ would be replaced by n ₀ for an n-type nanowire. Equation (3.12) states that λ _TF can be interpreted as a measure for the distance between charge carriers (related to the charge carrier density) in the nanowire under no applied bias. Therefore this parameter can be used to simulate the effect of the back gate in an experimental setup. The presented linearized model is not capable of describing nonlinear effects such as inversion mode of operation. However, the model distinguishes between accumulation/depletion mode of operation by allowing to choose between an n- or p-type material and different values of λ _TF.

3.3.6 Debye Screening Length

Buffer characteristics influence device performance, as described for experimental [39, 51, 52] and predicted [32] conductance signals. The screening of the analyte signal by the electrolyte is implemented through (3.8) and (3.10), which depend on the Debye length:

$$\displaystyle{ \lambda _{\mathrm{D}} = \sqrt{ \frac{\epsilon _{0 } \epsilon _{3 } k_{\mathrm{B} } T} {2N_{\mathrm{A}}{e}^{2}I}} }$$

(3.13)

where ε ₀ is the free space dielectric constant, K is the Boltzmann constant, T is the absolute temperature of the experiment, N _A is Avogadro’s number, and I is the ionic strength of the solution. The expression for the ionic strength is given by:

$$\displaystyle{ I = 1/2\sum _{i}c_{i}z_{i}^{2} }$$

(3.14)

where c _i indicates the concentration of ion species i and z _i is its formal charge.

This description of the electrolyte assumes that (a) the electrolyte is in equilibrium, i.e. the chemical potential is at a minimum and (b) the value for λ _D used in (3.8) and (3.10) is equal to the Debye length of the bulk buffer. We note that in principle these values can differ due to the biofunctionalization layer [31].

3.3.7 Biofunctionalization Layer

The biofunctionalization layer represents the chemical modifications to the nanowire surface, enabling it to capture the desired analyte. It can be visualized as a forest of linker molecules, Fig. 3.1. These molecules are not densely packed and are supposed to be immersed in the buffer solution. For this reason the biofunctionalization layer is implemented solely as a distance parameter, providing a measure of the spatial extension of the linker molecules. If the linker molecules would be so densely packed to exclude the buffer, an extra dielectric constant would be necessary to describe the effect of the biofunctionalization layer on the signal. Charges on the linker molecules are not considered influencing the signal, but part of the background conductance, see Sect. 3.2.3. In principle the surface functionality of the nanowire is not uniform [6] and requires a combined description of the pH-dependent charge on the linker molecules as well as the oxide where a common description of the charge of the oxide layer is through the site-binding model [53].

3.3.8 Protein Charges

The coverage of the biofet with analyte is described in Sect. 3.3.2. A common usage for biofets is the sensing of proteins [3, 6–8]. BioFET-SIM is specifically tailored for this. The PROPKA method [54–56] is employed to compute the pK_a values of each ionizable amino acid of a given protein structure. The computed pK_a values are used to evaluate the protonation state (charge) of each ionizable residue i as a function of pH as:

$$\displaystyle{ q_{i}(\mathrm{pH}) = \frac{1{0}^{\mathrm{pK}_{\mathrm{a}}^{i}-\mathrm{pH} }} {1 + 1{0}^{\mathrm{pK}_{\mathrm{a}}^{i}-\mathrm{pH}}} - p(i) }$$

(3.15)

where p(i) = 1 for i ∈ {Asp, Glu, C-, Tyr, Cys} and p(i) = 0 otherwise. In (3.15), q _i (pH) is the probability for the amino acid to be protonated [57]. In the single charge model (Sect. 3.2.3) the computed charges are summed together and a single charge is placed at the center of mass of the protein (Fig. 3.3a). In the multiple charges model (Sect. 3.2.4) a three-dimensional charge distribution is obtained by placing the charge computed with (3.15) at the average of the coordinates of the terminal atoms of the side chain of residue i (Fig. 3.3b). It is assumed that protein binding to the nanowire does not affect the pK_a values computed by PROPKA, nor disrupt the overall protein conformation.

3.3.9 Summary

To summarize, the BioFET-SIM model evaluates the conductance sensitivity signal Δ G∕G ₀ (Sect. 3.2.2) of a biofet as depicted in Fig. 3.2. The model considers a nanowire of length L and radius r _nw and described through its permittivity (ε ₁), mobility (μ), charge carrier density (n ₀∕p ₀), and Thomas–Fermi screening length (λ _TF, Sect. 3.2.5). The nanowire is coated with an oxide layer of thickness r _ox and characterized by permittivity ε ₂. On top of the oxide layer a biofunctionalization layer (Sect. 3.2.7) of thickness r _bf is responsible for capturing the desired analyte. The analyte charges are evaluated according to the buffer pH (Sect. 3.2.8) and placed at a certain distance l from the nanowire. The conductance sensitivity is evaluated with the analyte charges or summed in a single charge (Sect. 3.2.3) or separate (Sect. 3.2.4). The entire system is immersed in a buffer solution described by the Debye screening length (λ _D, Sect. 3.2.6) and its permittivity (ε ₃).

3.4 Usage

The BioFET-SIM method is available as a graphical web interface at the address www.biofetsim.org where a link to an instruction video to its usage is present. BioFET-SIM is distributed as open source through GitHub, and links to the source code as well as to the command line version of the program are also present on the web interface page. The web interface appears as shown in Fig. 3.4. A detailed description of the web interface and how to use it, along with a description of the uses of the command line version, are given in [34]. The following sections give a brief description of the interface and how the number of sensed analyte is computed. In the last section a simple example of the interface usage is given.

Fig. 3.4

BioFET-SIM web interface at www.biofetsim.org. The screenshot shows the interface being used to simulate the pH response to the sensing of 4,000 avidin proteins using default nanowire values

3.4.1 Interface Usage

The upper part of the web interface permits the selection of the protein structure that will act as analyte. The structure can be simply defined through its PDB identifier, to be downloaded from the PDB database (www.pdb.org [58]), or uploaded in PDB format. A pH value for the charges evaluation (Sect. 3.2.8) is also inserted, with default value 7.4.

By initializing the interface, the chosen structure is displayed over a representation of the nanowire surface. In the interface it is possible to regulate the relative orientation of the protein structure to the surface.

In the lower part of the interface it is possible to define all the various parameters needed to characterize the biofet, as previously described in Sects. 3.2.1–3.2.7. The interface permits also to define the number of sensed analyte molecules, as described in Sect. 3.3.2.

Upon performing a BioFET-SIM calculation, the computed value for the sensitivity is displayed, along with a graph showing the change in sensitivity against a chosen simulation parameter. An example is given in Sect. 3.3.3. Through the interface it is possible to save a state file which records the chosen relative orientation of the protein and the nanowire surface and an input file for the command line version of the program. The command line version is a powerful tool that allows the user to reproduce a calculation as well as performing many similar ones changing only one parameter at the time.

3.4.2 Analyte Surface Coverage

In order to apply (3.7) or (3.9) the sensed charge density σ _b has to be defined. Once defined the geometry of the nanowire, the charge density can be retrieved from the number of sensed analyte molecules. The web interface has two possible options for inserting the number of sensed molecules: full coverage or user specified.

The full coverage option assumes that the analyte molecules are perfectly packed in order to completely fill the available surface of the biofunctionalization layer. Full coverage of the nanowire has been demonstrated experimentally [59]. Once the relative orientation of the protein to the surface is defined, the interface computes the area on the surface occupied by the molecule. The surface of the biofunctionalization layer is then divided by this area to compute the number of sensed proteins. See [34] for further details.

The user can also specify a fixed number of bound proteins. This can be used, e.g., when it is known that the number of linker molecules on the biofet surface remains constant, even if the proteins bind with different orientations.

It is important to note that this treatment of the protein—biofunctionalization binding assumes that all proteins bind with the same orientation to the surface.

3.4.3 Example: Sensitivity Dependence on Oxide Layer Thickness

In this section we show the basic usage of BioFET-SIM to simulate the effects of different thickness values of the oxide layer. The simulation uses the default values for a p-type doped silicon nanowire as found on the web interface and reported in Table 3.1. The conductance sensitivity is evaluated for the sensing of 4,000 avidin proteins (PDB identifier 1AVD) at pH 7.4 and the single charge model.

Table 3.1 Parameters defining the simulated biofet

The computed sensitivity is reported in Fig. 3.5. Avidin carries a positive charge and induces a negative signal on the p-type doped nanowire. As expected, to a thicker oxide layer corresponds a diminished (in absolute value) sensitivity. The web interface permits to obtain the plot of Fig. 3.5 with a few simple clicks.

Fig. 3.5

Computed sensitivity response to the sensing of 4,000 molecules of avidin when changing the oxide layer thickness from 1.0 to 5.0 nm

3.5 Applications

The following sections illustrate some of the possible applications of BioFET-SIM. Whenever possible, the computed results are compared to experimental results.

3.5.1 Single Charge Model

For some applications the simpler single charge model (Sect. 3.2.3) is sufficient. In particular the sensing of avidin (PDB identifier 1AVD [60]) or streptavidin (PDB identifier 1STP [61]) is nearly independent from the used model. See Fig. 4 of [33]. In fact these two proteins are quite symmetrical and do not present any orientation that would favor one type of charges over the other. Another situation when the single charge model is required is represented by the binding of proteins with random orientations (i.e., generic adsorption) over the biofet surface. Random orientations average the protein charge distribution to the protein overall charge.

Fig. 3.6

Computed sensitivity response to the sensing of 4,000 molecules of avidin (dotted line, square points) or streptavidin (full line, round points) on a biotinylated surface with the single charge model. Dependence of the signal on the buffer pH

3.5.1.1 Effect of pH on Signal

At a pH value of 7.4 (physiological pH) avidin and streptavidin have opposite charge. This difference was exploited in [3], where the two proteins could be discerned by their opposite signal. The computed difference in signal for the sensing of avidin and streptavidin is illustrated in Fig. 3.6. At pH 7.4 (vertical dashed line) PROPKA computes a charge of 16.6 and −9.4 e for avidin and streptavidin, respectively (Sect. 3.2.8). This results in a positive signal for streptavidin and a negative signal for avidin.

In the same experiment [3] Stern et al. recorded also the change in signal at different pH values when sensing avidin. Figure 3.6 (dotted line) shows how the expected signal changes with different pH. At low pH values a negative signal is expected. At pH values around 10 the predicted signal is close to zero, while at higher pH values it is possible to obtain a positive signal. Obviously, this simulation does not consider the possible effects of very acid or very basic environments on the structure of the protein [39].

3.5.1.2 Signal Dependence on the Debye Screening Length

The intensity of a signal depends also on the ionic strength of the employed buffer solution. At short Debye lengths (high ionic strength, Sect. 3.2.6) a small signal is expected. In [51] Stern et al. analyze the dependence of signal upon dilution of buffer. Figure 3.7 depicts the BioFET-SIM evaluated signal for different Debye lengths for avidin and streptavidin at pH 7.4. It is evident that at longer Debye lengths the difference between the two proteins signals is more and more accentuated. A more complex example on how to use the signal dependence on λ _D to extrapolate information on the system can be found in [4, 34] and in Sect. 3.4.3 (Fig. 3.11).

Fig. 3.7

Computed sensitivity response to the sensing of 4,000 molecules of avidin (dotted line, square points) or streptavidin (full line, round points) on a biotinylated surface with the single charge model. Dependence of the signal on the buffer Debye screening length

3.5.1.3 Signal Dependence on Analyte Concentration

Stern et al. in [3] also analyze how the signal changes for different concentrations of analyte streptavidin. The dependence of the number of captured analyte molecules on the bulk analyte concentration is needed to evaluate the corresponding signal. We present here a simple model to evaluate it. A more complex model can be found in [19], but the qualitative results are equivalent. We note that the values used by Nair et al. for the forward and reverse reaction constants in [18, 19] are typical for the binding of single strand DNA but not necessarily for (strept)avidin–biotin complexes.

In the following we consider the sensing reaction as at its equilibrium. The generic sensing of streptavidin on a biotinylated surface can be expressed as:

$$\displaystyle{ A + R \rightleftharpoons AR }$$

(3.16)

where A is the analyte (streptavidin), R the receptor and A R the complex. The equilibrium constant K, inverse of the dissociation constant K _D, is given by:

$$\displaystyle{ K = \frac{[AR]} {[A][R]} }$$

(3.17)

For the streptavidin–biotin complex a dissociation constant $K_{\mathrm{D}} = 4 \times 1{0}^{-14}$ M has been found [62] and consequently $K = 1/K_{\mathrm{D}} = 2.5 \times 1{0}^{13}$ M⁻¹.

The concentration of free analyte [A] in equilibrium with the complex [A R] with respect to the starting concentration of the added sample [A _tot] is simply expressed by:

$$\displaystyle{ [A] = [A_{\mathrm{tot}}] - [AR] }$$

(3.18)

and similarly for the number of free receptors:

$$\displaystyle{ [R] = [R_{\mathrm{tot}}] - [AR] }$$

(3.19)

Inserting (3.18) and (3.19) in (3.17):

$$\displaystyle{ K = \frac{[AR]} {([A_{\mathrm{tot}}] - [AR])([R_{\mathrm{tot}}] - [AR])} }$$

(3.20)

and using the substitutions $m = 1 + K([A_{\mathrm{tot}}] + [R_{\mathrm{tot}}])$ and K′ = K[A _tot][R _tot] it is possible to write the equation:

$$\displaystyle{ K{[AR]}^{2} - m[AR] + K^\prime = 0 }$$

(3.21)

with solutions

$$\displaystyle\begin{array}{rcl} [AR]& =& \frac{1} {2K}\left (m + \sqrt{{m}^{2 } - 4KK^\prime}\right ) \\ & =& \frac{1} {2K}\left (m -\sqrt{{m}^{2 } - 4KK^\prime}\right ){}\end{array}$$

(3.22)

of which only the solution with the minus sign is significant (the other one giving [A R] > [R _tot], which is not possible). Equation (3.22) defines the concentration of analyte-receptor complexes given the starting concentration of analyte in the sample [A _tot], the concentration of initial receptor molecules [R _tot], and the equilibrium constant K.

BioFET-SIM requires the number of conjugated molecules, that is the number of formed complexes. This number can be obtained as:

$$\displaystyle{ n_{AR} = [AR]\;V _{\mathrm{rec}}\;N_{\mathrm{A}} }$$

(3.23)

where V _rec is the total volume occupied by the conjugated analyte-receptor molecules. Supposing complete coverage by streptavidin molecules, the total volume V _rec occupied by the sensed molecules can be visualized as the outer shell in Fig. 3.8. The analyte occupied layer is supposed to be as thick as the analyte molecule itself (r _stp). The occupied volume can be computed as difference between the outer and inner volumes:

$$\displaystyle\begin{array}{rcl} V _{\mathrm{out}}& =& \pi {\left (r_{\mathrm{nw}} + r_{\mathrm{ox}} + r_{\mathrm{bf}} + r_{\mathrm{stp}}\right )}^{2}L \\ V _{\mathrm{in}}& =& \pi {\left (r_{\mathrm{nw}} + r_{\mathrm{ox}} + r_{\mathrm{bf}}\right )}^{2}L \\ V _{\mathrm{rec}}& =& V _{\mathrm{out}} - V _{\mathrm{in}} \\ & =& \pi \;r_{\mathrm{stp}}\left [2\left (r_{\mathrm{nw}} + r_{\mathrm{ox}} + r_{\mathrm{bf}}\right ) + r_{\mathrm{stp}}\right ]L{}\end{array}$$

(3.24)

Fig. 3.8

Graphical representation of a generic nano BioFET sensing streptavidin. Supposing full coverage, the analyte creates a layer around the nanowire with thickness r _stp. Each streptavidin molecule occupies one cuboid as illustrated

Alternatively, supposing that each streptavidin molecule occupies a cube of volume r _stp ³, as shown in Fig. 3.8, it is possible to express V _rec as:

$$\displaystyle{ V _{\mathrm{rec}} = n_{R_{\mathrm{tot}}}\;r_{\mathrm{stp}}^{3} }$$

(3.25)

where $n_{R_{\mathrm{tot}}}$ is the total number of effective receptor molecules. Using (3.25) it is possible to define [R _tot] as:

$$\displaystyle\begin{array}{rcl} [R_{\mathrm{tot}}]& =& \frac{n_{R_{\mathrm{tot}}}/N_{\mathrm{A}}} {V _{\mathrm{rec}}} \\ & =& \frac{n_{R_{\mathrm{tot}}}/N_{\mathrm{A}}} {n_{R_{\mathrm{tot}}}\;r_{\mathrm{stp}}^{3}} \\ & =& \frac{1} {r_{\mathrm{stp}}^{3}\;N_{\mathrm{A}}}{}\end{array}$$

(3.26)

The value obtained with (3.26) and the literature value for K are inserted in (3.22). The concentration of sensed molecules [A R] is so defined as a function of the starting sample analyte concentration [A _tot]. [A R] is inserted in (3.23) together with the value obtained with (3.24), to finally obtain the number of sensed analyte molecules n _{A R} to be used as input for BioFET-SIM.

The simulation was performed using the default values of BioFET-SIM (Table 3.1) together with those reported in Table 3.2. The obtained values for the sensitivity are reported in Table 3.3 and plotted in Fig. 3.9. Given experimental conditions as those reported in Tables 3.1 and 3.3, the simulation states that it is possible to expect a quasi-linear behavior in a concentration range 1–10 mM and a sensitivity limit of 0.1 mM. A different setup would be necessary in order to sense smaller concentrations of analyte, e.g. a larger nanowire radius to increase the number of captured molecules. Since sensitivity is inversely proportional to the nanowire radius [32], possibly a longer Thomas–Fermi screening length would be necessary to balance out the effects on sensitivity. It is important to remember that there are many simplifications behind the data of Table 3.3, but the qualitative result of Fig. 3.9 is anyway reliable. Further details can be found in [63].

Table 3.2 Parameters used to simulate the sensing of streptavidin (1STP) depending on the analyte concentration

Table 3.3 Streptavidin sensing simulation data

Fig. 3.9

Simulated signal response for the sensing of streptavidin in the concentration range 1 × 10⁻⁵ to 1 M with default BioFET-SIM settings. The dots correspond to the data of Table 3.3, while the dashed lines represent a sigmoid fit to guide the eye

3.5.2 Multiple Charges Model

When the sensed analyte has an asymmetric charge distribution, the single charge model is insufficient to describe its relative signal. However, since the underlying model is based on a linear equation it is possible to obtain the signal from distributions involving multiple charges as a sum of the individual responses. In the following sections we show the importance of the multiple charges model (Sect. 3.2.4) through the sensing of a model peptide and a more complex example employing nucleocapsid protein.

3.5.2.1 Model Peptide Sensing

In this section we simulate the sensing of a model peptide constituted by two protonated lysine residues at the N-terminus, an eight alanine residues bridge and two deprotonated aspartic acid residues at the C-terminus: KK8ADD. The model was built as a linear peptide using PyMOL [64]. At pH 7.4 PROPKA computes a charge of 0.22 e. The peptide has a clear asymmetric charges distribution: three positive charges at the N-terminus and three negative charges at the C-terminus and overall charge nearly zero. The single charge model would predict a nearly zero signal for the sensing of this peptide. If there was the possibility of a biofunctionalization layer capable of specifically capturing aspartic or lysine residues, it would be possible to selectively bind the peptide as represented in Fig. 3.10a, c, respectively. The corresponding simulated signal as function of the Debye screening length is shown in Fig. 3.10d. If otherwise the peptide would be simply adsorbed on the nanowire surface in a horizontal position, the expected signal would be nearly zero. The same zero signal would also be expected if the charges distribution would average out because of random capturing. In fact, in case of random capturing of the analyte no specific direction relative to the nanowire surface would be preferred. In this case the single charge model would be sufficient.

3.5.2.2 Nucleocapsid Protein Sensing

The sensing of nucleocapsid protein (N-protein), as performed by Ishikawa et al. [48], is presented in this section, to exemplify the potential of BioFET-SIM and the multiple charges model.

In their work Ishikawa and co-workers employed In₂O₃ n-type doped nanowire-based sensors. Table 3.4 reports the different from the default parameter values used to simulate such devices. At pH 7.4 PROPKA computes a charge of 6.4 e for N-protein. The sensing of this protein by an n-type doped nanowire-based sensor should produce a positive signal, but a negative one was recorded [48]. It is possible to use BioFET-SIM to investigate the possible causes of this unexpected signal.

Fig. 3.10

Possible orientations and corresponding signal for the model peptide KK8ADD. The BioFET-SIM simulation was conducted with default values and the multiple charges model. The capturing of the peptide can be specific for (a) Asp or (c) Lys or (b) the peptide could be simply adsorbed horizontally on the surface. (d) The simulated signals as function of the Debye screening length for the different peptide—nanowire relative orientations. Figures originally from [34]: (a)–(c) doi:10.1371/journal.pone.0045379.g004, (d) doi:10.1371/journal.pone.0045379.g005

N-protein has an asymmetric charge distribution, as shown in Fig. 3.11a, e in the case of pH = 7.4. The protein could be captured by the biofunctionalization layer as reported in Fig. 3.11a, where the bulk of the protein rich in negative charges is closer to the nanowire surface. We refer to this situation as up. It is also possible that the protein is captured as depicted in Fig. 3.11e, that is with the positive charges rich hairpin closer to the nanowire surface. We refer to this second possibility as down. The two possible orientations up and down would generate quite different signals at pH = 7.4, as depicted in Fig. 3.11c, g, respectively. In the up orientation a small negative signal is expected for Debye screening lengths up to 3 nm (Fig. 3.11c). In the down orientation a clear positive signal is expected at any buffer dilution (Fig. 3.11g).

Varying the pH of the buffer solution would change the expected signal. At pH  = 3.8 N-protein has a computed charge of 22.3 e, ca. three times higher than at pH = 7.4. The protein is now mainly constituted by positive charges and both orientations produce similar positive signals (Fig. 3.11b, f). At pH = 11.0, instead, N-protein has a computed negative charge of −3.2 e. The predicted signal for the up orientation is clearly negative at any buffer dilution (Fig. 3.11d). For the down orientation a nearly zero but positive signal is predicted for Debye screening lengths from 1 to 3 nm (Fig. 3.11h). It is interesting to note that, by inverting the charge of N-protein by changing the pH value from 7.4 to 11.0, also the expected signals for the up and down orientations are inverted.

Table 3.4 Parameters defining the simulated In₂O₃ n-type semiconducting nanowire-based device that differs from those of Table 3.1

Fig. 3.11

N-protein sensing. The protein can be captured in (a) up or (e) down orientations. Different signals when varying the Debye screening length from 1 to 3 nm are expected at the different pH values 3.8, 7.4, and 11.0: (b), (c), and (d), respectively, for the up orientation and (f), (g), and (h), respectively, for the down orientation

The different simulated signals for the sensing of N-protein show that the BioFET-SIM model with multiple charges can be employed to study the possible effects of different orientations of the analyte with respect to the nanowire surface, coupled with the effects of different pH values of the buffer solution. In the specific case of N-protein we showed that, given certain conditions (e.g. Fig. 3.11c), an unexpected negative signal is possible.

3.5.3 Antigen Sensing on an Antibody Functionalized Surface

In this section we present a complex application of BioFET-SIM. Following [4], we decided to study the sensing of an antigen captured by an antibody functionalized biofet surface. Differences in the surface functionalization led to different signals, even if stemming from sensing the same antigen in otherwise similar conditions. This is indicative of different average distances between the sensed antigen and the biofet surface. Antibody binding to the surface was made possible through either an N-terminus located on the antigen-binding fragment (Fab) or by one or both C-termini at the antibody base, as shown in Fig. 3.12. N-terminus binding led to a reported antigen—surface distance of 5. 9 ± 0. 6 nm. C-termini binding corresponded to an antigen—surface distance of 8. 4 ± 0. 4 nm [4].

We decided to study the effect of possible antibody orientations on the signal. We further developed the formalism presented in Sect. 3.2 in order to obtain a quantity (the relative sensitivity factor) which depends solely on Debye length and distance of the sensed analyte from the surface. The derivation of this quantity is described in Sect. 3.4.3.1. Section 3.4.3.2 describes the necessary steps to prepare the antibody and antigen structures for the BioFET-SIM simulations. Section 3.4.3.3 shows the different orientations of the antibody, and consequently of the sensed antigen, over the biofet surface that we considered in this study. Finally, by performing a series of simulations at different Debye lengths, corresponding to those employed experimentally in [4], it was possible to obtain through a fit the average distance of the sensed antigen from the nanowire surface. These results are presented in Sect. 3.4.3.4.

Fig. 3.12

The employed custom-prepared antibody–antigen complex. Different chains are represented with different colors. Moreover, the C-termini at the antibody base are highlighted in gray and the N-termini on Fab in brown. Figure originally from [34]: doi:10.1371/journal.pone.0045379.g006

3.5.3.1 Theoretical Background

The interface was employed using the multiple charges formalism to take in consideration the different charges of the antigen and the possible effect on the simulated signal of their different orientations. However given the complexity of the studied system the following data treatment was performed through the single charge model formalism. In the following equations l stands for the average distance of the sensed charges from the nanowire surface.

As described in Sects. 3.2.1 and 3.2.3, the base conductance and conductance sensitivity can be described through (3.2) and (3.6). Since the physical and geometrical properties of the nanowire are fixed throughout all the following simulations, G ₀ can be considered as constant. It is then possible to express the change in conductivity as:

$$\displaystyle{ \Delta G = K\varGamma \,\varGamma _{l}\,\sigma _{\mathrm{b}} }$$

(3.27)

where K collects all constant values. We define λ _D ^max as the value of the buffer Debye length that we consider as maximum dilution. The maximum change in conductivity at this value is:

$$\displaystyle{ \Delta {G}^{\max } = {K\varGamma }^{\max }\,\varGamma _{l}^{\max }\,\sigma _{ \mathrm{b}} }$$

(3.28)

where we assume that changing the buffer dilution does not affect the number of adsorbed analyte molecules even at maximum dilution, hence σ _b is the same as in (3.27). Considering a highly diluted buffer, i.e. λ _D ^max ≫ l, and substituting in (3.8) we can express Γ _l ^max as:

$$\displaystyle{ \varGamma _{l}^{\max } \simeq 2 \frac{r_{\mathrm{nw}}} {r_{\mathrm{nw}} + l}{\left (1 + \sqrt{ \frac{r_{\mathrm{nw } } } {r_{\mathrm{nw}} + l}}\right )}^{-1} }$$

(3.29)

The ratio between the change in conductivity at a given Debye length and the maximum possible value is:

$$\displaystyle{ \frac{\Delta G} {\Delta {G}^{\max }} = \frac{K\varGamma \,\varGamma _{l}\,\sigma _{\mathrm{b}}} {{K\varGamma }^{\max }\,\varGamma _{l}^{\max }\,\sigma _{\mathrm{b}}} ={ \frac{\varGamma \,\varGamma _{l}} {\varGamma }^{\max }\,\varGamma _{l}^{\max }} }$$

(3.30)

and after reordering it is possible to obtain:

$$\displaystyle{ \frac{\Delta {G\,\varGamma }^{\max }} {\Delta {G}^{\max }\,\varGamma } = \frac{\varGamma _{l}} {\varGamma _{l}^{\max }} }$$

(3.31)

After inserting the explicit expressions, we obtain

$$\displaystyle{ \frac{\varGamma _{l}} {\varGamma _{l}^{\max }} = \frac{2 \frac{r_{\mathrm{nw}}} {r_{\mathrm{nw}}+l}{\left (1 + \sqrt{ \frac{r_{\mathrm{nw } } } {r_{\mathrm{nw}}+l}}\exp (l/\lambda _{\mathrm{D}})\right )}^{-1}} {2 \frac{r_{\mathrm{nw}}} {r_{\mathrm{nw}}+l}{\left (1 + \sqrt{ \frac{r_{\mathrm{nw } } } {r_{\mathrm{nw}}+l}}\right )}^{-1}} }$$

(3.32)

where we define Γ _l ∕Γ _l ^max as the relative sensitivity factor. According to (3.32) the relative sensitivity factor depends solely on the Debye length of the simulated buffer and the average distance of the sensed charges, once chosen a value for the nanowire radius.

Using (3.5) it is possible to compute the values of Γ for different Debye lengths, including Γ ^max for λ _D ^max = 1, 000 nm. In practice, the command line version of BioFET-SIM was employed to obtain these values. The BioFET-SIM parameters were set as in Table 3.1. With these parameters we obtained G ₀ = 279. 0 nS.

Using BioFET-SIM one obtains the value of Δ G (and Δ G ^max) simply by multiplying $\frac{\Delta G} {G_{0}}$ with G ₀. It is then possible to plot the l.h.s. of (3.31) for different values of λ _D. This plot is to be fitted to the r.h.s. of (3.32) where l is the fitting parameter. From a series of measures at different Debye lengths, the average distance l of the sensed charge from the nanowire surface is obtained.

3.5.3.2 Antibody and Antigen Preparation

A suitable complex structure of a generic antibody and an antigen as used by Vacic et al. [4] was prepared. Only few full antibody structures have been resolved by X-ray crystallography. We used the structure of an intact IgG2a monoclonal antibody, ascension code 1IGT [65]. For the antigen we used the structure of the SEA domain of human mucin 1, with ascension code MUC1 [66].

The antigen structure was rigidly docked to the antigen-binding site of the antibody through the program AutoDock [67] and visually checked with the program PyMol [64] for a reasonable docking. The scope of this docking was only to obtain a feasible complex structure. The antibody–antigen complex is shown in Fig. 3.12.

Fig. 3.13

Different orientations of the antibody with respect to the nanowire surface were considered in this study. The antigen is indicated by a red arc. The antibody base is indicated by a double line. The point of binding between the antibody and biofet surface is indicated by a small circle. In orientations A, C, and G the antibody is bound through the C-termini. In orientations B, D, E, and F the antibody is bound through the N-terminus. Figure originally from [34]: doi:10.1371/journal.pone.0045379.g007

Since interested in sensing only the antigen, we made the antibody structure as neutral as possible. All positions in the antibody sequence were changed to glycine using PyMol. Only the side chains were changed, while the backbone spatial disposition was maintained as in the original structure. When BioFET-SIM computes the charges of a protein it assigns a positive charge to the N termini and a negative one to the C termini, according to the PROPKA results. For this reason we changed the four chains N termini to aspartic acids and the C termini to arginines, in order to counterbalance these charges. By using a complex between the neutral antibody and the chosen antigen, we ensured that the simulated signal originated only from the antigen charges.

3.5.3.3 Antibody Orientations

Thanks to the graphical web interface, different orientations of the antibody–antigen complex were tested, simulating binding through the C- or N-termini. The seven orientations that were taken in consideration in this study are reported in Fig. 3.13. Orientations A to C correspond to those considered in [4], while orientations D to G were specifically conceived for this study. The antibody is supposedly bound to the biofet surface through the C-termini in orientations A, C, and G. Conversely, in orientations B, D, E, and F binding is through the N-terminus. C-termini binding restricts the possible movement of the antibody with respect to the nanowire surface (the antibody remains upright), while binding through the Fab N-terminus allows more freedom of movement (the antibody can nearly lie down on the surface).

3.5.3.4 Antigen Sensing

Different orientations of the neutral antibody–antigen complex were tested, as reported in Fig. 3.13. A biofunctionalization layer of 0.5 and 1.0 nm was added for C- and N-termini binding to the nanowire surface, respectively. We considered a pH of 7.4 and kept fixed the number of proteins to 4,000. The raw data for the BioFET-SIM results for all the orientations for the values of Debye length employed by Vacic et al. (3.07, 9.7 and 30.7 nm) plus λ _D ^max = 1, 000 nm, together with the computed data for the relative sensitivity factor are reported in the supporting material of [34] (doi:10.1371/journal.pone.0045379.s005). Here we present in Fig. 3.14 the plot of the computed relative sensitivity factor for the different orientations, together with the fitting parameters.

Among the orientations relative to C-termini binding, orientation G (l _G = 9. 8 nm) results in an average antigen—surface distance most similar to the reported experimental value (8.4 nm). For N-terminus binding, orientations E and F (l _E = 7. 9, l _F = 5. 9 nm) give an average distance close to the experimental value (5.9 nm). Thus it is possible to postulate that an upright or slightly inclined orientation is preferred for C-termini binding, while when binding through one of the Fab N-termini the antibody is likely to be nearly flat on the biofet surface. Moreover, an orientation like C is quite unlikely (l _G = 9. 8 nm), as opposed to what suggested in [4].

Fig. 3.14

Relative sensitivity factors for the different orientations depending on the Debye length. The computed fitting parameters l are also reported for each orientation

3.6 Conclusions

BioFET-SIM is a simple but yet powerful tool for the qualitative evaluation of nanowire-based biosensor signals. Its main characteristic is a web interface that permits any user to set up a calculation with a few mouse clicks. Most of all, the graphical interface permits to explore the effect of different orientations of the sensed analyte on the simulated signal. The graphical interface also prepares input files for the command line version of BioFET-SIM, which in turn permits to easily run batch simulations varying any given parameter.

In this chapter we showed the theoretical basis for BioFET-SIM, both in its single charge (Sect. 3.2.3) and multiple charges (Sect. 3.2.4) implementation. This was followed by an explanation of basic usage of the method (Sect. 3.3), where we showed how to study the possible effects of changing a device parameter like the oxide layer thickness on the biofet signal (Sect. 3.3.3).

Among the possible applications of BioFET-SIM we showed in Sects. 3.4.1.1 and 3.4.1.2 the effect on the sensor signal of changes of the buffer solution pH or Debye screening length. As a notable application of BioFET-SIM we mentioned the possibility of simulating a qualitative calibration curve for the signal upon sensing an analyte in different concentrations (Sect. 3.4.1.3). In Sect. 3.4.2.2 we showed the importance of using a multiple charges description of an analyte when different relative orientations of the analyte with respect to the nanowire surface are possible. We demonstrated how, through BioFET-SIM, it is possible to predict the possible signal for different orientations at different buffer pH values. Finally, in Sect. 3.4.3 we showed a complex application of BioFET-SIM aimed at interpreting and rationalizing experimental data. We demonstrated how it is possible to use the graphical web interface to prepare a series of different orientations of an antibody–antigen complex. Consequently the computed data let us postulate which of the considered orientations are those most likely to happen.