7.6 Toy model of protein folding A four-residue protein can take on the four dif

Question

7.6 Toy model of protein folding A four-residue protein can take on the four different conformations shown in Figure 7.30. Three conformations are open and have energy ( > 0) and one is compact, and has energy zero. (a) At temperature T, what is the probability, po. of finding the molecule in an open conformation? What is the probability, pc, that it is compact? (b) What happens to the probability pc, calculated in (a), in the limit of very large and very low temperatures. (c) What is the average energy of the molecule at temperature T? open open open compact Figure 7.30: Toy model of protein folding showing four different conformations. (Adapted from K. Dill and S. Bromberg, Molecular Driving Forces, 2nd ed. Garland Science, 2011.)

Explanation / Answer

A...At the given temperature T, the weight of the three open state is = exp(-e) and have enery >0.

The weight of the compact state is 1 at energy is zero.

Hence the partition function Z = weight o number of compact structure of protein + weight of the number of the open conformation

weight o number of compact structure of protein= 1

weight of the number of the open conformation = 3 exp(-e)

Z = 1 + 3exp(-e)

Hence the probability of finding the protein in the open state are,

Po = 3 exp(-e) / Z

The probability of finding the protein in the closed stage is,

Pc = 1 / Z

B.... at the high temperature level the molecules become exited and protein become open. Its conformation is changed as at high temperature protein becomes denatured. Hence the energy of all the 4 open state is become >1

hence the energy of Pc is = ¼.

At low temperature the Pc molecules are at lowest energy state hence Pc = 1

C..... at the temperature T, the average energy U of molecules =

      = ePo + OPc (ePo is the number of molecules of open state, OPc is number of molecules of conformation state

     = 3e exp(-e) / 1 + 3 exp(-e)

    Hence U = 3e /1

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