<br>To think about the power of stabilization by circularization,  

we used HP (Hydrophobic-Polar) model. HP model is a kind of simplified protein  

folding model and in the model, protein chain is given as zig-zag stick on 2D lattice.  

Each residue has the characteristic H or P (Hydrophobic or Polar). In this model, if an H  

residue is next to another H residue except the case it is due to the covalent bond, it decrease

free energy because of hydrophobic interaction. In our model, the decreased energy by each hydrophobic  

interaction is defined as -E<span class="sitatuki">H</span>. We added another characteristic SAPs into this model.  

Through thinking about this model, we can simply think about the effect of SAPs reflected as the effect to possibility

to fold into native conformation. We thought SAPs interaction is so strong, so in the case we add SAPs at N terminus and  

C terminus, both ends are set next to each other in the model. So, let's think about the simplest model.

<br>The simplest model is the model with the number of residue is 4 and the sequence is HPPH. In this case, without SAPs, the number of folding is 4, excluding enantiomers and rotamers. The possible states and the energy are listed below.

<img src="図１" width="300px" height="auto" alt="Fig_1"></td>  

<td style="border-style: none"; align="center"><h2>Fig. 1

</h2></td>

<br>Only the first one is stable and its energy is -E<span class="sitatuki">H</span>. Because it's most stable, so we thought it's native state. The possibility to fold as native state is below.

 

<img src="式１" width="300px" height="auto" alt="Formula_1"></td>  

<br>To calculate this, we used canonical ensemble from statistical mechanics. The possibility of causing state <span class="italic">i</span> is calculated through the function below.

<img src="式2" width="300px" height="auto" alt="Formula_2"></td>  

<br>But with SAPs, the number of folding is 1 and the structure is the stablest one.  

 

<img src="図2" width="300px" height="auto" alt="Fig_2"></td>  

<td style="border-style: none"; align="center"><h2>Fig. 2

</h2></td>

<br>The possibility to fold native conformation is of course 1.

<img src="式3" width="300px" height="auto" alt="Formula_3"></td>  

<br>Compared with both models, we can obviously think that thanks to the addition of SAPs, we can increase the possibility to fold correctly; the stability of native structure is definitely increased.

 

<img src="式4" width="300px" height="auto" alt="Formula_4"></td>  

<br>Let's think about more complicated case. The number of residue is 6 and the sequence is HPPHPH. The possible foldings are shown below.

 

<img src="図3" width="300px" height="auto" alt="Fig_3"></td>  

<td style="border-style: none"; align="center"><h2>Fig. 3

</h2></td>

<br>As we did in the simplest model, we thought the most stable state is the native state; native state has -2E<span class="sitatuki">H</span> as its energy. In this case, the possibility to fold as native structure is below.

<img src="式5" width="300px" height="auto" alt="Formula_5"></td>  

<br>With SAPs, the possible structures are shown below.

<img src="図4" width="300px" height="auto" alt="Fig_4"></td>  

<td style="border-style: none"; align="center"><h2>Fig. 4

</h2></td>

<br>The possibility to fold as native structure is below.

 

<img src="式5" width="300px" height="auto" alt="Formula_5"></td>  

<br>As we have shown in the simplest case, by the addition of SAPs, the possibility to fold correctly is definitely increased.

 

<img src="式6" width="300px" height="auto" alt="Formula_6"></td>  

<br>However, we should be careful about SAPs' characteristic; SAPs can limit the structure by circularization, but of course, if the stabilized structure is different from native structure, the addition of SAPs means the increase of the stability of the denatured structure. This fact can be shown in the model. If we add SAPs to the ends of HPHPHHPPPHHH model, the most stable structure is changed.

 

<img src="図5" width="300px" height="auto" alt="Fig_5"></td>  

<td style="border-style: none"; align="center"><h2>Fig. 5

</h2></td>

<br>As shown above, native state's free energy is -5E<span class="sitatuki">H</span>, but stabilized structure's lowest energy is only -3E<span class="sitatuki">H</span>. This means that if we want to stabilize protein of interest with circularization using SAPs, we have to add linkers carefully to make their ends closed without breaking their native structure.

 

<br>[1] Dill K.A. (1985). "Theory for the folding and stability of globular proteins". Biochemistry. 24(6): 1501?9. doi:10.1021/bi00327a032. PMID 3986190.

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