E1: Orders of magnitude

Week 5. Due Wed 10 Feb 23:59, reviewed Thu 11 Feb

About 2 hours. Submit one PDF (handwritten and scanned is fine) or a notebook on Canvas. You may use an LLM, but be ready to explain any answer at the board on Thursday.

1. Nelson

Nelson, Biological Physics: Your Turn 1a, Problems 1.3, 2.2 and 2.5.

The software and web sites named in Problems 2.2 and 2.5 are from 2002 and mostly gone. Use these instead:

  • 2.2: open each structure in the browser at https://www.rcsb.org/3d-view/<code>, for example rcsb.org/3d-view/1MBN. This is the Mol* viewer. To measure a distance, turn on selection mode, click two atoms, and open Measurements. For the rhinovirus (4rhv), choose the biological assembly, not the asymmetric unit.
  • 2.5: skip part (a); the micelle file is no longer online. Instead, estimate the radius of a spherical micelle of 65 dodecyl (C12) surfactants. Each tail has volume 0.35 nm³ and is at most 1.67 nm long. For part (b), download the 128-lipid DPPC bilayer (dppc.tar.gz) from the CHARMM-GUI lipid archive and open dppc_n128.pdb in the Mol* viewer (Open Files). Measure the bilayer thickness between the phosphorus atoms of the two leaflets. The simulation box is 6.4 nm × 6.4 nm, with 64 lipids per leaflet: what is the area per lipid?

2. The thermal energy scale

Calculate \(k_BT\) at body temperature (37 °C) in J, eV, kJ/mol and pN·nm. Compare it with the energy of one ATP hydrolysis (about 20 \(k_BT\) in the cell) and with a covalent C–C bond (about 350 kJ/mol). What does the comparison tell you about which structures in the cell are stable against thermal motion?

3. Python warm-up: fit a line with an uncertainty

Generate synthetic data \(y = a t + b + \text{noise}\) with \(a = 2\), \(b = 0.5\), 30 points for \(t\in[0,10]\) and Gaussian noise with standard deviation 1. Fit a straight line and report \(a \pm \delta a\). Repeat 1000 times with new noise and compare the spread of the fitted \(a\) with the \(\delta a\) from a single fit. Plot one data set with the fit.

Hint: np.polyfit(t, y, 1, cov=True) returns the covariance matrix of the parameters.