Introduction

Protein interaction is widespread in all live units, and its importance is self-evident. In our design, red light can stimulate the combination of N-hrluc-PhyB and C-hrluc-PIF3 and then produce blue light to guide microalgae to move(Fig.1).Therefore, whether our design is reasonable and the experiment successes depends on the degree of combination of N-hrluc-PhyB and C-hrluc-PIF3. Based on chemical thermodynamic formula, the relationship between binding rate and concentration and affinity constant is established in two steps. In addition, as a commonly used light-controlled polymeric protein actuator, this model can not only guide our own experiment, but also provide a quantitative model for the field of optogenetics.

Design

The molecular reaction in the solution can be divided into two steps:the first step is red light stimulation of PhyB and PIF3 binding, the second step is that the binding of PhyB and PIF3 facilitates the binding of two parts(NHrluc、Chrluc) of the Sea kidney luciferase. The molecular reactions carried out in our model and solution are also divided into two steps.

1.1 The first part: only the binding of PhyB protein and PIF3 protein is considered.

In order to find out the effect of the initial concentration of PhyB protein and PIF3 protein on the binding rate of the two proteins, we considered the binding of only PhyB and PIF3 proteins, and through of two kinds of proteins .Then,we get a specific equation between the initial concentration of protein and the binding rate of the two proteins.

1.1.1  Parameter

Symbol	Meaning
A	Concentration of PhyB
B	Concentration of PIF3
Ka	Affinity constant
Kd	Dissociation constant
η	Initial concentration of PhyB
A0	Initial concentration of PhyB
B0	Initial concentration of PIF3
AB	Concentration of PhyB-PIF3

1.1.2  Hypothesis:

(1) After the binding of the two proteins, the solution contains three substances: protein PhyB, protein PIF3 and the combination of the two. That is, the material is conserved and there is no protein loss before and after the reaction.
(2)   Except for the initial protein concentration, other factors have the same effect on the binding of the two proteins.

1.1.3   Modeling

Considering the binding of PhyB protein and PIF3 protein in solution, there are/p>

If the free concentration of PhyB is A, the free concentration of PIF3 is B, and the concentration of PIF3 is AB, the binding constant is:

The dissociation constant is

Then the expression of the binding rate is

Available jointly by (2) (3) (4):

Because we assume that the initial concentration of PhyB protein is higher than that of PIF3 protein, that is,

Available jointly by (5)(6):

The specific equation between the initial concentration of the two proteins and the binding rate of the two proteins was obtained.Available from (7)

When the light with the wavelength of 650 nm and a frequency of 20 Hz, we know Kd=500nM [1]（fig.1）：

The main results are as follows: (1) similar to the barrel effect, the binding rate depends on the low concentration of protein, and it can be estimated from the diagram that the minimum threshold of the concentration of the two proteins is mol /L at the same time.

(2)If the combination rate of A and B is not considered under red light (dark), then

Under the excitation of red light,the value is drastically reduced due to the change of in the configuration of the protein ,only when,The combination rate in the dark can be ignored.

1.2 The second part: the combination of N-hrluc and C-hrluc.

Under the action of "optical switch", the distance between N-hrluc and C-hrluc becomes closer, which is equivalent to increasing the concentration. The effect of "optical switch" on protein binding is explored by using the definition of equivalent concentration to replace the "optical switch".

1.2.1   Parameter

Symbol	Meaning
Bi,i=1,2	Represent N-hrluc and C-hrluc protein parts respectively.
Ai,i=1,2	Represent PhyB and PIF3 respectively.
d	Distance between B1 and B2

1.2.2  Hypothesis:

(1) The addition of optical switch has no effect on the conformational and properties of the original protein.
(2)   After the optical switch binds, the original protein is pulled closer, that is, the distance is smaller.

1.2.3   Modeling:

As shown in the following figure, optical switches are added to N-hrluc and C-hrluc proteins .Under the irradiation of light,the optical switches begin to combine.However, due to the conformational structure of B1 and B2 does not change, so the affinity constant is unchanged, but the distance of B1and B2 is close, which is equivalent to the increase of concentration.

The equivalent concentration of a particle can be obtained as follows:

The volume can be defined by the spacing between proteins, that is,

So:

According to the size of the protein, in the least ideal case, the distance between N-hrluc and C-hrluc is 16 nm. Assuming that the spacing is 15 nm, 10 nm and 5 nm, the equivalent concentrations are mol/L、 mol/L、mol/L。

As long as the affinity constant of the two parts of fluorescein enzyme is less than ,it can be considered to be 100% binding.

1.3 Conclusion:

our system is mainly limited by optical switches, so the Chlorella vulgaris need to be able to express enough PhyB and PIF3.

1.4 Sensitivity analysis:

If we do not conduct plasmid, add sea kidney luciferase and sea kidney fluorescein to the culture medium directly, and using the red light stimulates the movement of microalgae, and determines the threshold of sea kidney luciferase, which is what we want to verify through experiments.

1.5 Prediction:

If mol/L 's sea kidney luciferase can make microalgae move, then according to our model, it is assumed that under the same light condition, the value under red light is much smaller than that under dark, and the affinity constant between the two parts of sea kidney fluorescein enzyme is less than , equivalent to simultaneous expression in microalgae mol/L N-hrluc-PhyB and C-hrluc-PIF of mol/L).

Introduction

After subtilisin-modified ribonuclease was split in 1958，multifarious proteins was split successfully certificated by experiments[1]. Renilla luciferase is split in our experiment. The two parts of the split Renilla luciferase need to be self-assembled under the guidance of the optical switch. Different split sites produce two different parts, and the assembly effect will be different. The split site of Renilla luciferase we choose is from the result of experiments by Paulmurugan R and Gambhir S S[2], which is time-consuming and laborious, so can we predict better sites with calculational methods? We have searched a large number of previous experimental results which hint some kind of regularities (Table.1). The split site is closely related to the secondary structure of the protein, and the good site is generally located in the loop region of the protein. In addition, the location of the split sites seem to be related to the factors such as Sequence conservation, Length of coil regions, and Center distance. Based on these rules we summarized, Protein-split Model is established to predict better protein-split positions. We chose nanoLuc protein in the iGEM library to verify our model. The combination of the two nanoLuc parts obtained by the split position of our model is better than their original one verified by our experiment that means our model is convincing.

Design

2.1 Three factors

We conclude that the protein-split sites may be related to three factors, Secondary structure (factor 1), Length of coil region(factor 2), Center distance(factor 3)(fig.1), and found 65 Splitting sites (table.1) by experiment.

2.2 BP neural network

We can use a typical BP neural network (Dichotomous model) to help us find good split sites. If we can find a suitable set of weights{w1, w2}, at the same time, the inputs (I1, I2, I3) = (Factor1, Factor2, Factor3), output(O1, O2)=(1,0), then the protein-split sites selection is actually solved. For any protein, we only need to download its secondary structure from the PDB database, each amino acid residue corresponds to a set of factors. As long as the factors are input into the neural network, a label will be output: T:1:[1, 0] ; F: 0: [0, 1] (Fig. 2). We find a set of appropriate weights {w1, w2} by training the sample to complete our model, and do the sensitivity and specification analysis.

2.3 Algorithm

See connected protein-split_supplement1.

2.4 Database

Input see connected protein-split_supplement2,Output see connected protein-split_supplement3.

2.5 Result

Peural network parameter

w1

1	2	3
-0.0306	0.233	0.5924
0.6037	-0.6431	-0.5429
1.3603	-0.6622	0.5412
0.6023	-0.3277	-0.5616

b1

0.7124	-0.464	-0.9178	-0.5768

w2

1	2
0.3485	-0.1632
-0.2581	0.1562
-0.8925	0.9408
-0.2699	-0.1891

b2

0.7532	0.1274

Performance Evaluation Criteria

2.6 Conclusion

Our model is highly sensitive, but have low specificity which produces high ratios of false positives, meaning that many bad protein-split sites are recognized as good protein-split sites(Fig.3). However, in general, good cleavage sites is much less than bad cleavage sites in a protein with hundreds of amino acid residues, so the low specificity is inevitable. We only need to ensure that very few good sites are not mistaken for bad sites. This model will provide several predicted good sites to instruct experiment, as long as there are one true good sites. Because our model has a very low false negative, this model is fabulous in practical applications. Even more incredible is that our model is successfully verified by experiment.

2.7 Experimental Verification

We chose nanoLuc in iGEM library to split to test our model. Based on protein-split model, we calculated an optimal cleavage site and predicted that the original cleavage site is not good enough(Fig.4). The experimental results show that the ability of the two parts of nanoLuc(st-1 and sc-1) cut at our predicted cleavage site to self-assemble is much greater than the original cleavage site(st-2 and sc-2)(Table.2 and Table.3).