- #1
Matt atkinson
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Homework Statement
A simple model of a polymer undergoing a helix-coil transition is to describe the polymer
in terms of N equal length segments, each of which can be in either a coil or
a helix state. A more realistic model also takes into account the energy cost associated
with a boundary between a helix section and a coil section which leads to
cooperativity. This is analogous to the Ising model.
Write a simple Monte Carlo simulation to model a polymer helix-coil transition. Each
of the N segments can be in a coil or helix state and can switch state with a probability
depending on the energy cost of switching state. You may assume that the change in
free energy of a segment switching from coil to helix in units of ##k_BT## is
$$\frac{∆Fh}
{k_BT}
= a(T − T_c)$$
where a is a constant, ##T## is the temperature and ##T_c## is the critical temperature for the
transition. Your simulation should compare generated pseudo random numbers with
the relevant probability to decide whether a segment should switch state or not. You
should start your simulation with an initial state for your polymer and then run over
many steps of trial segment state switches. You will need enough steps so that the
polymer has relaxed to equilibrium. Repeat your simulation experiment many times to
get good averages. Run your simulation at different temperatures to find the fraction
of helices as a function of temperature.
It is recommended to do the case with no cooperativity first. To introduce cooperativity,
i.e. a non-zero energy cost of junctions between segments of different states,
you will need to modify your code to check the state of a segment’s neighbours to
calculate the probability of switching state.
Homework Equations
The Attempt at a Solution
Just confused as to if I am approaching the problem correctly;
In c++ I set up at array of size Nx1 where N is the number of monomers, or elements.
I then have relevant code to flip the particle form a "helix" to "coil" state if the probability given by ##e^{\Delta E}## where for the non-cooperativity would be just ##e^{-a(T-T_c)}## right? this part I understand. Also if ##\Delta E<0## or the probability given above is less than a generated random number then the state change will occur because it minizes entropy.
In my simulation I have used the value T as ##T-T_c## because no specific polymer was given and i range the value of T from -10 to 10. The constant "a" is apparently a scale factor to fit with experimental data.
Here's my problem, i have no idea how to add cooperativity maybe some local energy of +1 or -1 if the adjacent elements are in the same or opposite state?
To get the fraction of helix's I looped a huge number of times over the flip probability and then add up the number of elements with the value I set for the for the "helix" state.
I can attach my code if that would help?
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