# Team:HokkaidoU Japan

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

In our project, we tried to stabilize protein structure by circularization using SAR. About the detailed concept of circularization, please read circularization page. To think about the power of stabilization by circularization using SAP, we used HP (Hydrophobic-Polar) model. HP model is a kind of simplified protein folding model and in this model, protein chain is given as zig-zag stick on 2D lattice. Each residue has the characteristic H or P (Hydrophobic or Polar). To calculate the stability of protein structures, in this model, if an H residue is next to another H residue without covalent bond, it decreases free energy because of hydrophobic interaction. In our model, the decreased energy by each hydrophobic interaction is defined as -EH. We added another characteristic SAR into this model. Through thinking about this model, we can simply think about the effect of SAR reflected as the effect to probability to fold as native state. We thought SAR interaction is so strong, so in the case we add SAR 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.

The simplest model is the model with the number of residue is 4 and the sequence is HPPH. In this case, without SAR, the number of states is 4, excluding enantiomers and rotamers. The possible states and the energy are listed below. KB is Boltzmann constant 1.38064852*10

Only the first one is stable and its energy is -EH. Because it's most stable, we thought it's native state. The probability to fold as native state (P

To calculate this, we used canonical ensemble from statistical mechanics. The probability of causing state i is calculated through the function below. Ei is the energy of state i, Ej is the energy of the state j.

But with SAR, the number of states is 1 and the state is the most stable one.

The possibility to fold native conformation (PHPPH

Compared with both models, we can obviously think that thanks to the addition of SAR, we can increase the probability to fold correctly; the stability of native state is definitely increased.

Let's think about more complicated case. The number of residue is 6 and the sequence is HPPHPH. The possible states are shown below. Also, we excluded enantiomers and rotamers.

As we did in the simplest model, we thought the most stable state is the native state; the native state has -2EH as its energy. In this case, the possibility to fold as native structure (PHPPHPH

With SAR, the possible states are shown below.

The probability to fold as native structure (PHPPHPH

As we have shown in the simplest case, by the addition of SAR, the probability to fold correctly is definitely increased.

As we have shown, circularization using SAR can stabilize protein native structure. However, we should be careful about SAR' characteristic; SAR can limit the structure by circularization, but of course, if the stabilized structure is different from native structure, the addition of SAR means that it increase the stability of the denatured structure. This can be shown in the model. If we add SAR to the ends of HPHPHHPPPHHH model, the most stable structure is changed.

As shown above, native state's free energy is -5EH, but stabilized structure's lowest energy is only -3EH. This means that if we want to stabilize a protein with circularization using SAR, we have to be careful about the difference between its native structure and stabilized structure. If they have huge difference, we have to add linkers not to break its native structure by circularization using SAR.

[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.

[2] Rob Philips. (2008) "Physical Biology Of the Cell". Garland Science

In our project, we tried to stabilize protein structure by circularization using SAR. About the detailed concept of circularization, please read circularization page. To think about the power of stabilization by circularization using SAP, we used HP (Hydrophobic-Polar) model. HP model is a kind of simplified protein folding model and in this model, protein chain is given as zig-zag stick on 2D lattice. Each residue has the characteristic H or P (Hydrophobic or Polar). To calculate the stability of protein structures, in this model, if an H residue is next to another H residue without covalent bond, it decreases free energy because of hydrophobic interaction. In our model, the decreased energy by each hydrophobic interaction is defined as -EH. We added another characteristic SAR into this model. Through thinking about this model, we can simply think about the effect of SAR reflected as the effect to probability to fold as native state. We thought SAR interaction is so strong, so in the case we add SAR 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.

The simplest model is the model with the number of residue is 4 and the sequence is HPPH. In this case, without SAR, the number of states is 4, excluding enantiomers and rotamers. The possible states and the energy are listed below. KB is Boltzmann constant 1.38064852*10

^{-23}(J/K), T is temperature (K).

Fig. 1 |

Only the first one is stable and its energy is -EH. Because it's most stable, we thought it's native state. The probability to fold as native state (P

^{wild}HPPH) is below.

To calculate this, we used canonical ensemble from statistical mechanics. The probability of causing state i is calculated through the function below. Ei is the energy of state i, E

But with SAR, the number of states is 1 and the state is the most stable one.

Fig. 2 |

The possibility to fold native conformation (PHPPH

^{SAR}) is of course 1.

Compared with both models, we can obviously think that thanks to the addition of SAR, we can increase the probability to fold correctly; the stability of native state is definitely increased.

Let's think about more complicated case. The number of residue is 6 and the sequence is HPPHPH. The possible states are shown below. Also, we excluded enantiomers and rotamers.

Fig. 3 |

As we did in the simplest model, we thought the most stable state is the native state; the native state has -2EH as its energy. In this case, the possibility to fold as native structure (PHPPHPH

^{wild}) is below.

With SAR, the possible states are shown below.

Fig. 4 |

The probability to fold as native structure (PHPPHPH

^{SAR}) is below.

As we have shown in the simplest case, by the addition of SAR, the probability to fold correctly is definitely increased.

As we have shown, circularization using SAR can stabilize protein native structure. However, we should be careful about SAR' characteristic; SAR can limit the structure by circularization, but of course, if the stabilized structure is different from native structure, the addition of SAR means that it increase the stability of the denatured structure. This can be shown in the model. If we add SAR to the ends of HPHPHHPPPHHH model, the most stable structure is changed.

Fig. 5 |

As shown above, native state's free energy is -5EH, but stabilized structure's lowest energy is only -3EH. This means that if we want to stabilize a protein with circularization using SAR, we have to be careful about the difference between its native structure and stabilized structure. If they have huge difference, we have to add linkers not to break its native structure by circularization using SAR.

[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.

[2] Rob Philips. (2008) "Physical Biology Of the Cell". Garland Science