# Uncertainty Estimation for Quantitative Agarose Gel Electrophoresis of Nucleic Acids

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

- (1)
- instrumental errors Δ
_{1}related to the physical realization of electrophoretic separation, quality of the agarose gel, the geometry of the well and during the placement of the sample (these sources of uncertainty cause both systematic bias in the measurement results and the variability of the results when performing multiple measurements); - (2)
- errors Δ
_{2}associated with the mathematical processing C = f(I) of images obtained as a result of electrophoretic separation. These sources of uncertainty are associated with errors caused by determining the boundaries of bands corresponding to certain lanes, with the incorrect correlation of intensity distribution along the well with the electrophoretic mobility scale, with procedures of baseline removal and separation of overlapping peaks. Here, C is the result of the substance quantity measurement, and I is the processed image.

_{1}are transformed when performing mathematical processing f, it is difficult to determine the distribution density for Δ in advance (since it is determined, among other things, by the sample under study; —the results of peak separation in the intensity distribution depend on the complexity of its composition). As a result, it is difficult to divide the error into systematic and random components, so that the limiting normalized values for them would be valid for any composition of nucleic acids subjected to separation.

## 4. Results

#### 4.1. Uncertainty Assessment in the Presence of Standard Samples of Separable Nucleic Acid Mixtures with Known Concentrations

- The electrophoretic separation system is calibrated for a sample of nucleic acids (provided that the electrophoretic mobility of the mixture components is different), whose concentration C
_{i}, i = 1, 2, …, k in the mixture is known with a relative error γ_{i}. When calibrating the electrophoretic system for this sample, n repeated measurements of the corresponding peak areas in the signal h(m) are performed, the results of which are denoted as ${\widehat{S}}_{ij}$, j = 1, 2, …, n. To eliminate random noise in the results of electrophoretic separation in the system used, a sensitivity threshold C_{min}is set, such that the contents of substances smaller than this threshold are not registered: if C < C_{min}, the measurement result S = 0.

- 2.
- Immediately after calibration, the sample substance to be measured is introduced into the electrophoretic separation system. The sensitivity threshold set during calibration and, consequently, the absolute systematic error ΔS caused by this threshold, remain unchanged. Electrophoretic separation is performed several times to estimate random errors. It is reasonable to take the number of measurements equal to n—as being performed during the calibration. Accordingly, in the i-th analysis, the values of the area ${S}_{ij}$ will be obtained:$${S}_{ij}=K\cdot {C}_{acti}-\Delta {S}_{i}+{\epsilon}_{ij}$$

_{i}and the coefficient of proportionality K do not change during the calibration and measurement.

- 3.
- Next, the statistical processing of the results of multiple measurements when calibrating the electrophoretic separation system and when using it to measure the composition of the sample of interest is performed.

- –
- mean values of the results of calibration and measurement$${\widehat{M}}_{i}=\frac{1}{n}\cdot {{\displaystyle \sum}}_{j=1}^{n}{\widehat{S}}_{ij},{M}_{i}=\frac{1}{n}\cdot {{\displaystyle \sum}}_{j=1}^{n}{S}_{ij},$$
- –
- estimates of random error variances$${\hat{\sigma}}_{i}^{2}=\frac{1}{n-1}\cdot {{\displaystyle \sum}}_{j=1}^{n}{({\widehat{S}}_{ij}-{\widehat{M}}_{i})}^{2},{\sigma}_{i}^{2}=\frac{1}{n-1}\cdot {{\displaystyle \sum}}_{j=1}^{n}{\left({S}_{ij}-{M}_{i}\right)}^{2},$$
- –
- estimates of the arithmetic means variances$${\hat{\sigma}}_{\mathrm{mean}i}^{2}=\frac{{\hat{\sigma}}_{i}^{2}}{n-1},{\sigma}_{\mathrm{mean}i}^{2}=\frac{{\sigma}_{i}^{2}}{n-1},$$
- –
- upper bounds ${\widehat{V}}_{\mathrm{max}i}$, ${V}_{\mathrm{max}i}$ of one-sided confidence intervals for variances of arithmetic mean with confidence probability P = 0.95 based on quantiles of ${\chi}^{2}$ distribution with the number of degrees of freedom equal to (n − 1)$${\widehat{V}}_{\mathrm{max}i}=\frac{n-1}{{\chi}_{1-P}^{2}\left(n-1\right)}\cdot {\hat{\sigma}}_{\mathrm{mean}i}^{2},{V}_{\mathrm{max}i}=\frac{n-1}{{\chi}_{1-P}^{2}\left(n-1\right)}\cdot {\sigma}_{\mathrm{mean}i}^{2}$$
- –
- estimates of variances of relative random errors of mean values$${\hat{\gamma}}_{\mathrm{mean}i}^{2}=\frac{{\widehat{V}}_{\mathrm{max}i}}{{\widehat{M}}_{i}^{2}}=\frac{n-1}{{\chi}_{1-P}^{2}\left(n-1\right)}\cdot \frac{{\hat{\sigma}}_{\mathrm{mean}i}^{2}}{{\widehat{M}}_{i}^{2}},{\gamma}_{\mathrm{mean}i}^{2}=\frac{{V}_{\mathrm{max}i}}{{M}_{i}^{2}}=\frac{n-1}{{\chi}_{1-P}^{2}\left(n-1\right)}\cdot \frac{{\sigma}_{\mathrm{mean}i}^{2}}{{\widehat{M}}_{i}^{2}}.$$

- 4.
- The content ${C}_{acti}$ of the i-th component of the test sample is calculated as the solution to the system of equations from steps 1 and 2 of this list:$$\{\begin{array}{c}{\widehat{M}}_{i}=K\cdot {C}_{i}-\Delta {S}_{i},\\ {M}_{i}=K\cdot {C}_{acti}-\Delta {S}_{i}.\end{array}$$$${C}_{acti}=\frac{{M}_{i}+\Delta {S}_{i}}{{\widehat{M}}_{i}+\Delta {S}_{i}}\cdot {C}_{i}\mathrm{or}{C}_{acti}={C}_{i}+\frac{{M}_{i}-{\widehat{M}}_{i}}{K}.$$
- 5.
- The expression for the relative error (uncertainty) of the results of measuring the quantity of a substance when using electrophoretic separation is the sum$$\frac{\Delta {C}_{acti}}{{C}_{acti}}=\frac{\Delta {\widehat{M}}_{i}}{{\widehat{M}}_{i}+\Delta {S}_{i}}+\frac{\Delta {M}_{i}}{{M}_{i}+\Delta {S}_{i}}+\frac{\Delta {C}_{i}}{{C}_{i}}\approx \frac{\Delta {\widehat{M}}_{i}}{{\widehat{M}}_{i}}+\frac{\Delta {M}_{i}}{{M}_{i}}+\frac{\Delta {C}_{i}}{{C}_{i}}$$$$\Delta {C}_{acti}=\Delta {C}_{i}+\frac{\Delta {M}_{i}-\Delta {\widehat{M}}_{i}}{K}-\frac{\left({M}_{i}-{\widehat{M}}_{i}\right)\cdot \Delta K}{{K}^{2}},$$$$\frac{\Delta {C}_{acti}}{{C}_{acti}}=\frac{\Delta {C}_{i}}{\frac{{M}_{i}+\Delta {S}_{i}}{{\hat{M}}_{i}+\Delta {S}_{i}}\xb7{C}_{i}}+\frac{\Delta {M}_{i}-\Delta {\hat{M}}_{i}}{\frac{{M}_{i}+\Delta {S}_{i}}{{\hat{M}}_{i}+\Delta {S}_{i}}\xb7{C}_{i}}-\frac{({M}_{i}-{\hat{M}}_{i})\xb7\Delta K}{\frac{{M}_{i}+\Delta {S}_{i}}{{\hat{M}}_{i}+\Delta {S}_{i}}\xb7{C}_{i}\xb7{K}^{2}}\approx \frac{\Delta {C}_{i}}{{C}_{i}}\xb7\frac{{M}_{i}+\Delta {S}_{i}}{{\hat{M}}_{i}+\Delta {S}_{i}}+\frac{1}{{C}_{i}}\xb7({\hat{M}}_{i}\xb7\frac{\Delta {M}_{i}}{{M}_{i}}-\frac{{\hat{M}}_{i}^{2}}{{M}_{i}}\xb7\frac{\Delta {\hat{M}}_{i}}{{\hat{M}}_{i}}-({M}_{i}-{\hat{M}}_{i})\xb7\frac{{\hat{M}}_{i}+\Delta {S}_{i}}{{M}_{i}+\Delta {S}_{i}}\xb7\frac{\Delta K}{K})$$

#### 4.2. Uncertainty Assessment for a Case Where There Are No Standard Samplesthe Case When There Are No Standard Samples of the Nucleic Acid Mixture to Be Separated with Known Concentrations

- –
- processing of the image of intensity distribution along the depth of the well in the agarose gel obtained during electrophoretic separation (alignment, rotation, removal of local geometric deformations, compensation of the background intensity in the processed image fragment);
- –
- a statistical integrated estimate of the intensity within each line of the well’s image and formation of signal h(m), where m is the value of the electrophoretic mobility corresponding to each line of the image;
- –
- removal of the underlying background and baseline h
_{0}(m) using the construction of the lower envelope for the signal h(m); - –
- applying of the iterative procedure for separating the overlapping peaks in the signal (h(m) − h
_{0}(m)).

_{1}, I

_{2}, … I

_{n}are compared with each other as follows. Their average I

_{mean}, obtained by pointwise averaging the obtained images’ sample, is considered. The alignment of the two images is performed by finding such horizontal and vertical (if required) offset, which provides the maximum of the mutual correlation function of the compared images:

_{1}, I

_{2}, … I

_{n}and estimating the error value S for a sample of the obtained results is a less good option because the algorithm f is significantly sensitive to the error value of the initial data (it contains procedures for solving incorrect problems). Their significant values can lead to distortions in assessing the underlying background and baseline, failure to detect hidden peaks, and so on. Since ${\sigma}_{\mathrm{mean}}^{2}$ is significantly less than the value of ${\sigma}^{2}$ at each point (x, y), the processing of image ${I}_{\mathrm{mean}}$ will be free from these drawbacks and, in addition, will require less time than the processing of all images I

_{1}, I

_{2}, … I

_{n}. Such circumstances lead to the necessity of processing a censored sample, for which the above-mentioned rule, such as Equation (5), is not always valid.

_{0}(m)). Since the processed spectrogram must contain strictly positive values, in practice, the underlying background is estimated by plotting the lower envelope of the signal h(m). The values (h(m) − h

_{0}(m)) at m, corresponding to the forbidden values of electrophoretic mobility, estimate the limit of possible systematic error ${\Delta}_{\mathrm{syst}}$(m).

_{0}(m)) as f

_{h}. Its result is the estimated content C of substance

_{act}(m)) and the impulse response function r(m) of the electrophoretic nucleic acid agarose gel separation system:

_{act}(m) is an ill-conditioned problem, known in mathematics as the problem of solving the Fredholm integral equation of the first kind.

_{act}(jω), where j is an imaginary unit, and ωω is a circular frequency. To avoid complex numbers, one must use one of the following techniques:

- –
- using symmetrization of the transformed signals by reflecting them from the semiaxis of real values m > 0 to the semiaxis of negative values of the argument m, i.e., h(–m) = h(m) and r(–m) = r(m), and using the cosine transform, leading to strictly real-value spectra, instead of the integral Fourier transform;
- –
- using the Hartley integral transform, —akin to the Fourier and the cosine transform.

_{act}(ω).

_{act}(m) should result in narrowing of the peaks and their separation, which is necessary to isolate the main peak responsible for the target yield of the transcription products and the side peaks responsible for the yield of side products on the spectrogram obtained from the electrophoretic separation instrument for nucleic acids in agarose gel. This problem is incorrect. The proposed solution uses the Tikhonov regularization method [32] reformulated as the minimal modulus principle [33], which is conveniently implemented in the domain of Fourier, cosine or Hartley transforms. However, its exact implementation leads to the Gibbs effect, which consists of the emergence of solution distortions in the form of oscillatory components in the recovered signal. To avoid this, we should apply a modification of the Howard algorithm [34], which leads to minimization of the negative values of h

_{act}(m). This procedure is an iterative process, in its original formulation, based on the use of forward and inverse Fourier transforms. In our case, one of the points of modification of the Howard algorithm is the use of transforms equivalent to Fourier transforms, which provide real-valued results (the already mentioned symmetrization and cosine transform or the Hartley transform). The procedures listed, which are the essence of algorithm f, keep the peak areas unchanged, which makes it possible to ensure the reliability of the results of measuring the concentration of substances in the sample analyzed using the electrophoretic separation of nucleic acids in agarose gel. In the presented formulation, however, no transformations involving an analytic continuation to the complex plane are used, which leaves the possibility of accompanying the calculation of f values with the automatic obtaining of values of partial derivatives of a given function and thereby automating the obtaining of estimates using expressions (5) and (6).

## 5. Discussion

- –
- the need to involve information about the accuracy of standard samples (if used);
- –
- the need to perform multiple measurements to achieve better accuracy (if they are possible);
- –
- the need for analysis of technical documentation to identify all available information on all components of instrument uncertainty for the instruments used;
- –
- the need to clearly distinguish between random and systematic components of the uncertainty budget for quantitative results derived from electrophoretic separation of nucleic acids in agarose gel;
- –
- when applying the approach based on numerical modeling of peak area estimation, a significant number of calculations are required (this can pose a challenge when automating electrophoresis).

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Complex-Step Method for Derivatives Estimation

**x**) from multiple real-valued arguments ${\mathrm{x}}^{T}=\left({x}_{1},{x}_{2},\dots ,{x}_{k}\right)$. If this function permits the analytical continuation to the complex plane for variable x

_{t}, then we can use the complex-step approximation $es{t}_{C}$ of partial derivative $\partial q\left(x\right)/\partial {x}_{t}$, which is much more accurate than finite difference estimate $es{t}_{R}=\frac{q\left(x+{d}_{t}\right)-q\left(x\right)}{\alpha}$ and can be calculated as $es{t}_{C}=\frac{\mathrm{Im}\left(q\left(x+j\cdot {d}_{t}\right)\right)}{\alpha}$. Here,

**d**is a vector with the same size with

_{t}**x**filled with zeros, except for its t-the element equal to some small number α~0, and $j=\sqrt{-1}$ is an imaginary unit.

_{t}(α = 10

^{−w}·x

_{t}, where w is usually equal to 10 ÷ 20) to prevent a situation where the derivative estimate will be equal to zero due to the difference (q(

**x**+

**d**)–q(

_{t}**x**)) being less than the machine precision allows to represent. This can lead to a significant final error. On the contrary, in computing $es{t}_{C}$, we can always take α equal to 10

^{−200}, and such decision will not cause errors in derivative estimation. The main reason is that we are not faced, in that case, with computationally unstable operations, such as the subtraction of two close numbers, as it takes place in finite difference calculation.

**x**) is real, so we end up with the following expression:

^{a}

^{×b}, where a and b are the dimensions of the processed image I, to the set C

^{a}

^{×b}of complex numbers. Since the function f reflects a computational algorithm, this continuation is possible. Therefore,

_{(x,y)}is a matrix of the same dimensions as image I, composed of zeros, except for the cell with indices (x, y), which contains one.

## Appendix B. The Monte Carlo Method for Estimating Uncertainty Bounds

- To calculate the value of the function C
_{0}= f(I) corresponding to the initial set I of argument values for f. - To generate N random combinations I
_{j}of argument values of the calculated function f; index j runs the values 1, 2,…, N; the generation is conducted according to a uniform distribution, since, in the general case, there is no reason to prefer some values from Ω over Ω to others in the absence of information about the function f. - To compute values C
_{j}= f(I_{j}). - To estimate the boundaries of interval $J\approx \left[{\mathrm{min}}_{j}{C}_{j},{\mathrm{max}}_{j}{C}_{j}\right]$.
- To estimate the marginal error of the value of function f caused by the inaccuracy of its arguments using the expression $\Delta C={\mathrm{max}}_{j}\left|{C}_{j}-{C}_{0}\right|$.

^{6}[35,48]. For the presented variant, N can be reduced to 10

^{3}[49]. The confidence probability P for the interval J and corresponding value ΔC can be assessed by the following expression [49]:

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## Share and Cite

**MDPI and ACS Style**

Semenov, K.; Taraskin, A.; Yurchenko, A.; Baranovskaya, I.; Purvinsh, L.; Gyulikhandanova, N.; Vasin, A.
Uncertainty Estimation for Quantitative Agarose Gel Electrophoresis of Nucleic Acids. *Sensors* **2023**, *23*, 1999.
https://doi.org/10.3390/s23041999

**AMA Style**

Semenov K, Taraskin A, Yurchenko A, Baranovskaya I, Purvinsh L, Gyulikhandanova N, Vasin A.
Uncertainty Estimation for Quantitative Agarose Gel Electrophoresis of Nucleic Acids. *Sensors*. 2023; 23(4):1999.
https://doi.org/10.3390/s23041999

**Chicago/Turabian Style**

Semenov, Konstantin, Aleksandr Taraskin, Alexandra Yurchenko, Irina Baranovskaya, Lada Purvinsh, Natalia Gyulikhandanova, and Andrey Vasin.
2023. "Uncertainty Estimation for Quantitative Agarose Gel Electrophoresis of Nucleic Acids" *Sensors* 23, no. 4: 1999.
https://doi.org/10.3390/s23041999