How long should it take the polypeptide backbone of a 6-residue, 10-residue, 15-residue and 20-residue folding nucleus to explore all its possible conformations? Assume that the polypeptide backbone randomly reorients every 10-13 seconds (s).
10-7s, 10-3s, 102s, 107s, respectively
Understanding the time it takes for a polypeptide backbone to explore all its possible three-dimensional shapes, or conformations, is a fundamental concept in protein folding, often illustrating the magnitude of the Levinthal paradox. This paradox highlights the immense number of possible conformations a polypeptide chain can adopt.
We are given the time it takes for the polypeptide backbone to randomly reorient itself and transition between conformations, which is $10^{-13}$ seconds (s).
We need to calculate the total time required to explore all possible conformations for polypeptides of lengths 6, 10, 15, and 20 residues.
Calculating Polypeptide Exploration Time
To determine the total time, we need to estimate the total number of possible conformations for each polypeptide length. A common simplification used in such estimations assumes that each amino acid residue can independently adopt a certain number of distinct backbone conformations (related to the $\phi$ and $\psi$ dihedral angles). Based on the provided options and the resulting calculations, it appears the underlying model assumes that an N-residue polypeptide has approximately $10^N$ possible backbone conformations.
The total time ($T$) to explore all conformations is the product of the number of conformations and the time taken for each reorientation step:
$$ T = (\text{Number of Conformations}) \times (\text{Time per Reorientation}) $$
Using the assumption that the number of conformations for an N-residue polypeptide is $10^N$ and the reorientation time is $10^{-13}$ s, the formula becomes:
$$ T(N) = 10^N \times 10^{-13} \text{ s} $$
Let's calculate the exploration time for each given polypeptide length:
- For N = 6 residues:
Number of conformations = $10^6$
Total time $T(6) = 10^6 \times 10^{-13} \text{ s} = 10^{6-13} \text{ s} = 10^{-7} \text{ s}$
- For N = 10 residues:
Number of conformations = $10^{10}$
Total time $T(10) = 10^{10} \times 10^{-13} \text{ s} = 10^{10-13} \text{ s} = 10^{-3} \text{ s}$
- For N = 15 residues:
Number of conformations = $10^{15}$
Total time $T(15) = 10^{15} \times 10^{-13} \text{ s} = 10^{15-13} \text{ s} = 10^{2} \text{ s}$
- For N = 20 residues:
Number of conformations = $10^{20}$
Total time $T(20) = 10^{20} \times 10^{-13} \text{ s} = 10^{20-13} \text{ s} = 10^{7} \text{ s}$
The calculated times for exploring all possible conformations for polypeptides of length 6, 10, 15, and 20 residues are $10^{-7}$ s, $10^{-3}$ s, $10^{2}$ s, and $10^{7}$ s, respectively.
| Polypeptide Length (N) | Assumed Number of Conformations | Time per Reorientation | Total Exploration Time |
|---|---|---|---|
| 6 residues | $10^6$ | $10^{-13}$ s | $10^{-7}$ s |
| 10 residues | $10^{10}$ | $10^{-13}$ s | $10^{-3}$ s |
| 15 residues | $10^{15}$ | $10^{-13}$ s | $10^{2}$ s |
| 20 residues | $10^{20}$ | $10^{-13}$ s | $10^{7}$ s |
These results indicate the exponentially increasing time required to sample the vast conformational space as the polypeptide chain length increases, a key aspect of the protein folding problem.
Important Questions from Biochemistry
The following table lists names of scientists and advances made by them
Column A Column B A Linus Pauling (i) Myoglobin structure B Emil Fischer (ii) Model of α-helix C John Kendrew (iii) Lock and Key model D Christian Anfinsen (iv) Sequence-structure
Which one of the following options correctly matches contents of column A with column B?One gram of a polysaccharide composed of 1000 glucose units has the same effect on osmolarity as that of
Several proteins are modified by phosphorylation at specific amino acid residues to alter their activities. Which one of the following amino acids is NOT typically a site of phosphorylation in proteins?
The following statements are made with regard to the optical activity of amino acids derived from natural proteins:
A. All alpha-amino acids have the D stereochemical configuration.
B. All L-amino acids have the (S) absolute configuration except cysteine, which has the (R) absolute configuration.
C. All D-amino acids have the (S) absolute configuration except cysteine, which has the (R) stereochemical configuration.
D. In the absolute configuration system, L-threonine and L-isoleucine are (2S, 3R)-threonine and (2S, 3S)-isoleucine diastereomers, respectively.
Which one of the following options represents the combination of all correct statements?
Lactose is the substrate for which of the following enzyme ?
