Introduction

The combination of N-hrluc-PhyB and C-hrluc-PIF3 stimulated by red light can produce blue light and cause microalgae to move.Therefore, the degree of combination of N-hrluc-PhyB and C-hrluc-PIF3 directly determines whether our design is reasonable and the experiment is successful. Based on the simplest chemical thermodynamic formula, the relationship between binding rate and concentration and affinity constant was established in two steps.In addition, as a commonly used photopolymeric protein actuator in photogenetics, our work can not only guide our own experiments, but also provide a simple quantitative model for this field.

Background

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 PIF6 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 PIF6 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 PIF6 protein in solution, there are/p>

If the free concentration of PhyB is A, the free concentration of PIF6 is B, and the concentration of PIF6 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 PIF6 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 PIF6.

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).