Theoretical Analysis Of Protein Folding Rate Limit

According to the transition state theory,the protein folding rate depends on the folding free energy barrier and a pre-exponential factor term,which sets a time scale across the energy barrier,that is,the folding rate limit.It is of great significance to explore the dependence of folding rate limit for understanding the mechanism of protein folding.Based on the experimental facts,two theoretical hypotheses are proposed to estimate the protein folding rate limit: one is that the protein folding satisfies the weak collision limit of Kramers rate theory,and the folding rate limit is proportional to the barrier height;the other is that the folding rate limit depends on the k-mer information of the reduced amino acid sequence.According to the first hypothesis,we estimate the height of the folding free energy barrier(n is the coupling vibration degree of freedom)with the power of 1/n of the natural state nonlocal contact number of protein.Based on Kramers escape rate theory in the case of weak collision limit,we obtain a prediction model of protein folding rate.In this model,the folding rate limit is proportional to the number of natural nonlocal contacts.The model was tested on an experimental data set containing 150 protein folding rates.When n=3,the model obtained a fitting degree of about 79% to the experimental values.The results show that the number of nonlocal contacts in the natural state of protein may be related to the protein folding rate limit,because if the folding rate limit is set as a constant and the height of the folding free energy barrier is estimated by the power of 1/n of the number of nonlocal contacts in the natural state,only 69% of the experimental value can be obtained.In addition,when n=1,the model degenerates to the topological isomer search model proposed by Makarov et al.According to the second hypothesis,we first divide 20 amino acids into two groups,and use the k-mer information of the reduced amino acid sequence to quantitatively describe the folding rate limit.We found that: 1)the k-mer information can be used as an index to explain the 10 fold rate difference among the five WW domains studied by Munoz et al.2)the k-mer information and the number of nonlocal contacts are used to estimate the pre-exponential and the height of free energy barrier respectively,and the goodness of fit of 150 proteins is 75%,which is about 6% higher than that of constant pre-exponential.In addition,if we reduce the amino acid sequence according to the degeneracy of each amino acid,we get similar results.These results indicated that the protein specificity of folding rate limit(pre-exponential factor)can be measured by the amount of short-range association information of amino acid sequence.