Problem
DNA is a polymer whose backbone contains deoxyribose sugar and phosphate groups. The phosphate groups carry negative charge, so DNA migrates toward the positive electrode in an agarose gel, while the gel matrix slows larger fragments more than smaller ones.
Assumption for sizing: within the ladder range, \(\log_{10}(\text{bp})\) depends approximately linearly on migration distance \(d\) (mm) measured from the wells.
| Band (bp) | Distance \(d\) (mm) | \(\log_{10}(\text{bp})\) |
|---|---|---|
| 2000 | 27 | \(\log_{10}(2000)=3.30103\) |
| 1500 | 32 | \(\log_{10}(1500)=3.17609\) |
| 1000 | 39 | \(\log_{10}(1000)=3.00000\) |
| 500 | 52 | \(\log_{10}(500)=2.69897\) |
| Unknown | 35 | ? |
Visualization (agarose gel lanes)
Step 1: Set up the regression model
Model the ladder with a straight line: \[ y = a + b\,d,\quad \text{where } y=\log_{10}(\text{bp}) \text{ and } d \text{ is migration distance (mm).} \]
After fitting \(a\) and \(b\) from the ladder, predict \(y\) for the unknown distance and convert back using \(\text{bp}=10^{y}\).
Step 2: Compute the slope \(b\) and intercept \(a\)
- Compute the means: \[ \bar{d}=\frac{27+32+39+52}{4}=37.5 \qquad \bar{y}=\frac{3.30103+3.17609+3.00000+2.69897}{4}=3.04402 \]
- Use the least-squares formulas: \[ b=\frac{\sum (d_i-\bar{d})(y_i-\bar{y})}{\sum (d_i-\bar{d})^2}, \qquad a=\bar{y}-b\,\bar{d} \]
- Compute the sums (rounded): \[ \sum (d_i-\bar{d})(y_i-\bar{y})\approx -8.49425, \qquad \sum (d_i-\bar{d})^2 = 353 \] \[ b\approx \frac{-8.49425}{353}=-0.024063 \]
- Compute the intercept: \[ a=\bar{y}-b\,\bar{d} =3.04402-(-0.024063\cdot 37.5) =3.94639 \]
Fitted standard curve:
\[ \log_{10}(\text{bp}) \approx 3.94639 - 0.024063\,d \]
Step 3: Predict the unknown fragment size at \(d=35\) mm
- Predict \(y\): \[ y_u = 3.94639 - 0.024063\cdot 35 = 3.10418 \]
- Convert back to base pairs: \[ \text{bp}_u = 10^{\,3.10418}\approx 1271 \]
Estimated size: \(\text{bp}_u \approx 1.27\times 10^3\) bp, i.e., about 1270 bp.
Reporting to two significant figures is common for gel-based sizing unless the gel, ladder, and measurement precision are high.
Biology note: why “deoxyribose sugar” matters on a gel
DNA’s deoxyribose sugar forms part of the repeating sugar–phosphate backbone. The phosphate groups (not the bases) provide a near-uniform negative charge density per unit length of DNA, so separation on agarose gel is primarily by fragment length (bp) rather than by sequence.