Photosynthetic formaldehyde 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. or 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 formaldehyde assimilation pathway

The metabolism of photosynthetic formaldehyde assimilation was shown on Fig.1. Since the substrate (Ru5P) and product (F6P) of the 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 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 formaldehyde (Chen et al., 2010). In this case, we utilize the mathematical principles described above to analyze the metabolism.

Fig.1 Schematic diagram of photosynthetic formaldehyde 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 formaldehyde 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 c are parameters.

(10)

Meaning of the parameters:
k1,k3 related to ATP and NADPH2;
k2 related to HPS/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.

Results

By changing the value of those parameters (k1, k2 … ), we obtained the relationship (Fig.3) between the concentration of different components (Ru5P, F6P and formaldehyde) 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 absorbed into the plant cell, the original steady state would be 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.

Link to other modeling

Reference

Chen, L. M., H. Yurimoto, K. Z. Li, I. Orita, M. Akita, N. Kato, Y. Sakai and K. Izui (2010). "Assimilation of formaldehyde in transgenic plants due to the introduction of the bacterial ribulose monophosphate pathway genes." Biosci Biotechnol Biochem 74(3): 627-635.

Song, Z., I. Orita, F. Yin, H. Yurimoto, N. Kato, Y. Sakai, K. Izui, K. Li and L. Chen (2010). "Overexpression of an HPS/PHI fusion enzyme from Mycobacterium gastri in chloroplasts of geranium enhances its ability to assimilate and phytoremediate formaldehyde." Biotechnol Lett 32(10): 1541-1548.