Photosynthetic HCHO assimilation pathway

Mathematical Principles

Almost all chemical reactions obey the law of constant proportion:

(1)

where vi(l) is the constant proportion of reaction component Ai in the l-th reaction. The components with vi(l)>0 are the resultants. On the contrary, they are reagents. Let ϛl represents the extent of l-th reaction which means the relevant components increase vi(l) mol when ϛl=1. Thus, the equation between ϛl and vi(l) can be given by:

(2)

ni(l) is the quantity of i-th component. Then reaction rate Jl can be defined by:

(3)

V is the reaction volume. Assume that the quantity of component Ai changes merely due to the chemical reaction and the exchange of outside and inside. Base on the assumption, the law of conservation of matter can be established for ni of every component Ai:

(4)

Xi is the concentration of i-th component. I or J stands for the exchange of outside and inside. Due to the changes of reaction system volume is very slight, the relative growth rate shall be thought equal to zero. Let for the concentration of Ai in the environment, the principle of dynamics of can be expressed by:

(5)

Finally, we obtain dynamic equation group of the reaction system:

(6)

Photosynthetic HCHO assimilation pathway

The metabolism of photosynthetic HCHO assimilation was shown on Fig.1. Since the substrate (Ru5P) and product (F6P) ofthe sequential reactions catalyzed by HPS and PHI are intermediates of the Calvin cycle in plants, photosynthesis could provide sufficient substrates for the reactions catalyzed by HPS and PHI if the two enzymes were expressed in plant(Song, Orita et al. 2010).It has been proved that over-expressing the HPS/PHI fusion protein can enhance the ability of the plants to absorb and assimilate exogenous HCHO(Chen, Yurimoto et al. 2010).In this case, we utilize the mathematical principles described above to analyze the metabolism.

Fig.1 Schematic diagram of photosynthetic HCHO assimilation pathway. Ru5P, D-ribulose 5-phosphate; Hu6P, D-arabino-3-hexulose 6-phosphate; F6P, fructose 6-phosphate; Xu5P, xylulose 5-phosphate; RuBP, ribulose 1,5-bisphosphate; 3-PGA, glycerate 3-phosphate; FBP, fructose-1,6-bisphosphatase;

Simplify the system (Fig.2), consist of the input of HCHO and CO2, the recycle of Ru5P and the output of F6P:

Fig.2 A simplified version of photosynthetic HCHO assimilation pathway

The mechanism of this system can be represented by a set of chemical reactions, as shown below.
(7)
a, b and their linear combination were the constant proportion of reactions. If a/b=m(or k1/k2∝m), then m represents the competition between J1 and J2. We know that a+b=3 (base on the traditional calvin cycle), then Equ.7 can transform to below and the reaction rate also shown in Equ.8:
(8)
Therefore, the dynamic equation group of the reaction system was obtained as shown in Equ.9:
(9)
We can simplify the function into an elegant form, a1, a2, b1, b2 and care parameters.
(10)

By changing the value of those parameters (k1, k2…), we obtained the relationship (Fig.3)between the concentration of different components (Ru5P, F6P and HCHO)versus the time.

Fig.3 The diagram of concentration versus time. A for Ru5P and F6P, B for HCHO and C for those three components
From Fig.3, we found that the components tend to be the steady state when time goes by.Means that when the formaldehyde into the plant cell, the original steady state was broken, but after a period of time, the cells will restore homeostasis which indicated that the indoor formaldehyde has been absorbed by the plant. For figure 3B, in the initial stage, formaldehydewill continue to grow due to the delayed effects of reaction; subsequently,the concentration of formaldehyde begins to decrease with time and finally tends to be the steady state.

Meaning of the parameters:
k1,k3 related to ATP and NADPH2;
k2related toHPS/PHI;
k4: the speed of F6P transferred to outside;
m:the competition between J1 and J2;
[F0]: the initial concentration of F6P;
[C0]: the initial concentration of formaldehyde.

Folate-independent pathway

The metabolism of folate-independent pathway was shown on Fig.4.

Fig.4 Schematic diagram offormaldehyde metabolic pathways in plant. SMM cycle, S-methylmethionine cycle
Simplify the system(Fig.5), consist of the input of HCHO and the recycle of HCOOH:
Fig.5 A simplified version of folate-independent pathway. FTS, 10-Formyl-THF synthetase.
Some research indicated that the activity of FTS intobacco was low and we ignored this pathway in our system. The chemical reactions of this formaldehyde metabolism are shown below.
(11)
where parametera is a constant which means the proportion of HCOOH into the calvin cycle. The dynamic equation group of the reaction system can be obtained:
(12)
By changing the value of the parameters (k0, k1…), we obtained the relationship (Fig.6) between the concentration of different components (HCOOH and HCHO) versus the time.

Fig.6 The diagram of concentration versus time.
We found that the components tend to be the steady state when time goes by. Means that when the formaldehyde into the plant cell, the original steady state was broken, but after a period of time, the cells will restore homeostasis.

Model of stoma

While formaldehyde diffused into plant cells through the stoma, it suffered a series of obstruction or resistance of diffusion. Subsequently, a range of complex chemical reactions activated when the formaldehyde in cell diffused intercellular space. By analyzing the process, the molecule of formaldehyde diffused into cell through the stoma, we found the reason of the happening process is the mass transportcaused by the uneven distribution of formaldehyde density. Therefore, the formaldehyde molecular diffusion velocity distribution meets the maxwell speed distribution function in normal temperature conditions. That means the number of formaldehyde encountered the unit leaf areawithin per unit time along the x-axis direction were obtained by the below equation.
(1)
where the vx, vyand vz are the components of velocity of the gas molecules on the x-axis, y-axis and z-axis, respectively. is the gas molecule maxwell speed distribution function. ? is the number of gas molecule diffused into unit leaf areawithin per unit time. S,t is the leaf area and time, respectively. In equation , the n stands for the number of formaldehyde in air, m for molecular weight of HCHO, k for boltzmann constant and T for temperature. From Equ.1, we got:
(2)
We knew that the average velocity of gas molecule in temperature Tcan be shown as:
(3)
substitute into Equ.2, we got:
(4)
Molecular mass of the formaldehyde from high concentration along the direction of diffusion of low concentrationwithinthe unit leaf area per unit time can be described by:
(5)
where ?b and ?a stand for the number of gas molecules inside stoma and outside stoma.△ρ for the density difference. We also assumed that the temperature and humidity are constants.The stomatal conductance (Gs) can be defined which the molecular mass diffused into plantwithinthe unit leaf area and per unit time.
(6)
According to hydrodynamics, the compressible gas exits the continuity equation showed below if the temperature and humidity remain unchanged.
(7)
where C stands for concentration. The unit of MC/St is μmolHCHO?m–2?s–1, stands for the nubmer of formaldehyde absorbed by the plantwithinthe unit leaf area and per unit time.Substitute into Equ.6:
(8)
We assumed parameter P stands for the net absorption rate for plant leaf:
(9)
The Pindicated the absorbing ability of formaldehyde. In our project, we assume P is a constant.η is the ratio of formaldehyde used in the reaction. Noted that if η=0, it means all the gases absorbed in plant do not participate in any chemical reaction. If η=1, it means all the gases participated in the chemical reaction and were absorbed by those reaction. Substitute into Equ.8, we obtained:
(10)
where C stands for the concentration difference inside and outside stoma. Therefore, Equ.10 is the relationship between stomatal conductance and net absorption rate of plant leaf, concentration difference inside and outside stoma when the temperature and humidity remain unchanged.It indicated that plant leaf stomatal conductance and net HCHO absorption rate is proportional and inversely proportional to the concentration difference.

In future work, we would like to clone the stomatal regulation gene, AtAHA2, to the expression vector.Due to its powerful function of enhancing the degree of stomatal opening, we assumed that the gene can improve the absorption efficiency of formaldehyde. Therefore, in transgenic tobacco, the parameter ηis larger than that in the wild type plants. We plotted the two situations and the diagram was shown in Fig.1.

Fig.1 The relationship between stomatal conductance and the concentration of formaldehyde in air for differ η. In the situation of same stomatal conductance, the concentration of formaldehyde in air for transgenic tobacco is less than that of wild type tobacco, which means the gene AtAHA2 can improve the absorption efficiency of formaldehyde.