20.181:hw11 sols

Solutions to HW 11

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Part 1:

To complete the code shell provided, you had to insert something like the following code snippet:

           tau_new = (a_i_old[rxn] / (a_i_new[rxn]+eps)) * (tau_i[rxn] - t_cur) + t_cur
           tau_i[rxn] = tau_new
           a_i_old[rxn] = a_i_new[rxn] 

note that you can't update a_i until after you've calculated a new tau.

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Part 2:

An easy way to solve this question is with dimensional analysis. If you've never heard of dimensional analysis before, it's essentially a fancy way of saying: manipulate the data to produce a value with the correct units; more often than not, this value will also be the correct quantity.

For instance, we know that we're trying to calculate the probability per unit time that a particle collision causes a reaction. Our solution should therefore be in units of [1/time] (probabilities are unitless!).

There is only one way to combine the given data so as to produce our desired units:

[math]\displaystyle{ \frac{1\times10^9}{M\cdot sec}\cdot \frac{1 \mbox{ } mole}{6.023\times10^{23}} \cdot \frac{1}{1 \times 10^{-15} \mbox{ } L} = 1.66 \mbox{ } sec^{-1} }[/math]

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Part 3:

Observe above a "switching event," where pA and pB "switch" steady states. Note that neither the original abundances of pA nor pB are steady state values - the proteins settle into their steady states towards the end of the simulation.

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Part 4:

See above :)