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.

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.

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

Chlamydomonas reinhardtii

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.

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

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".