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S3 bio (Numbers (1 animal cell \(\sim10\mu\text{m}\), Filament lengths…
S3 bio
Stat mechanics
The probability that a system occupies a microstate \(s\)
\(p_s=\frac{1}{Z}e^{-E_s/kT}\)
where the partition function (or weight)
\(Z = \sum_i e^{-E_i/kT}\)
over \(i\) microstates where \(E_i\) is the total energy
Receptor-ligand binding
We have \(L\) ligands in a box with \(\Omega\) lattice sites
We take \(\varepsilon_b<\varepsilon_{\text{sol}}\), i.e. energy is lowered by RL binding
For all unbound ligands we have \(E_T=L\varepsilon_\text{sol}\) and a multiplicity of
\(\omega=\frac{\Omega !}{L!(\Omega - L)!}\approx \frac{\Omega^L}{L!}\)
possible arrangements. \(\omega\) holds for indistinguishable items (i.e. two ligands can be swapped and not count as unique states)
As all arrangements have equal energy, we can take
\(Z=\omega e^{-L\varepsilon_\text{sol}/kT}\)
For one bound ligand we have \(E_T=(L-1)\varepsilon_\text{sol}+\varepsilon_b\) and a multiplicity of
\(\omega=\frac{\Omega !}{(L-1)!(\Omega - L+1)!}\approx \frac{\Omega^{L-1}}{(L-1)!}\)
And similarly for \(Z\)
The probability that a ligand is bound is given by the ratio of weights
\(p_\text{bound}=\frac{\sum_\text{states}\text{1 bound}}{\sum_\text{states}\text{0 bound}+\sum_\text{states}\text{1 bound}}\)
By solution we can write \(p\) as a function of ligand concentration \(c=L/V_\text{box}\) with reference \(c_0=\Omega/V_\text{box}\)
\(p_\text{bound}=\frac{(c/c_0)e^{-\beta\Delta\varepsilon}}{1+(c/c_0)e^{-\beta\Delta\varepsilon}}\)
where \(\Delta\varepsilon =\varepsilon_b-\varepsilon_\text{sol}\)
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Numbers
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1 bacterium \(\sim1\mu\text{m}\)
Swim speed \(\sim10\mu\text{m}\,\text{s}^{-1}\)
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bioelectricity
Use \(kT\sim 4.3\cdot 10^{-21}\)J for muscle cells at \(\sim 37^{\,\circ}\)
Membrane potential is the PD created by a concentration gradient of charged ions
If the membrane is not permeable to some other ions, the permeable ions may not distribute evenly either side
In equilibrium (no ion movement) the membrane potential is equal to the Nernst potential and the membrane is in Donnan equlibrium
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