Copper ion sensor detection involves two transduction pathways: copper ion metabolism and S. cerevisiae cell response to pheromone[1]. First, the activated pCUP1 promoter acts as a sensor detector to recognize copper ions in the cytoplasm and guide the transcription and translation of α-factor gene into pheromone protein. The generated pheromone acts as input signal to a signal converter. Second, G protein receptor in the signal converter binds to the dissociative pheromone so as to initiate MAPK signal pathway[2][3] to express fluorescent protein. Finally, the cultured strains are sent to the flow cytometry to detect fluorescence intensity, and then the detection result is displayed on the display screen.

1.1 Copper ion metabolism model

As a gene regulatory element that can recognize environmental signals inside and outside a cell, the activated pCUP1 promoter initiates the transcription process of target gene[4]. The process of copper ion metabolism satisfies the Michaelis-Menten condition[5], so a system of differential equations is used to describe the process of pCUP1 driving α-factor gene to generate pheromone

where
x-- Copper ion concentration
s1 -- Concentration of pheromone mRNA generated during transcription
s2 -- Concentration of pheromone generated during translation
γ1 -- Basal expression level of α-factor gene
β1 -- Maximum transcription rate of α-factor gene
Kx-- Binding constant of copper ion
μ1-- Degradation rate of pheromone mRNA
τ1 -- Translation rate of pheromone mRNA
μ2--Degradation rate of pheromone

1.2 MAPK signal pathway model

As shown in Fig. 1, Pheromones generated during copper ion metabolism are recognized and bound by specific G protein receptors on the surface of S. cerevisiae cell[6]. These pheromones coupling to G protein-coupled receptor (GPCR) triggers the dissociation of heterotrimeric G protein (Gαβγ), and the released Gβγ dimer activates the MAPK cascade through a key intermediate Ste5. Next, MAP kinase activates GFP gene transcription after acting on Ste12. Considering Ste12 as intermediate node, we model the process of S. cerevisiae cell response to pheromone in two stages.

Figure 1 MAPK signal pathway in S. cerevisiae cell

(1) Upstream MAPK cascade

Assume that [Fus3] is the concentration of activated Fus3, [Ste120] is the concentration of Ste12 unphosphorylated by Fus3, vSte12 is the rate of Ste12 phosphorylated by kinase Fus3, and vFus3 is the reaction rate of Fus3. To indicate the positive correlation between vSte12 and vFus3, a correlation equation is established by

If [Ste12p] denotes the concentration of Ste12 phosphorylated by Fus3, then the change rate of [Ste12p] satisfies a first-order reaction equation[7]

where λ is the rate of [Ste12p] dephosphorylation. Substituting (3) into (4), we obtain

The steady-state condition for the above reaction process requires

Using the conservation law

Eq. (6) can be converted to

where [Ste12T] is the total concentration of Ste12. Thus, the proportion of Ste12 phosphorylated by Fus3 is formulated as

Because the process of Ste12 phosphorylated by Fus3 is in a black box, the effect of the input variable [Fus3] on the reaction model is mainly discussed. Denoting an intermediate variable μ = vFus3/λ and an increasing saturation function , we simplify Eq. (7) as follows

where . Similar to Eq. (10), the phosphorylation of Ste11 and Ste7 by Fus3 can be described by

In Fig. 2, G protein is activated to recruit Ste11 so that the concentration of activated G protein (i.e., G*) is in one-to-one correspondence with . Thus, the phosphorylation process in (11) can be formulated by replacing with G*

According to Hill function[8], we have

and then

where GT is the total concentration of G proteins and K2 is the Mie constant of pheromone.

Figure 2 MAPK cascade of protein kinases.

(2) Downstream protein kinase network

Assuming that y (a.u.) is the expression intensity of GFP and s3 (μM) is the concentration of GFP mRAN, a system of differential equations is used to model protein kinase network

where μy is the degradation rate of GFP, and γ2, β2, μ3 and βy are the basal transcription rate, maximal expression level, degradation rate and translation rate of GFP mRNA, respectively.

1.3 Transfer function of copper ion detection

The entry of copper ions into S. cerevisiae cell to participate in metabolism will cause changes in the intracellular metabolic pathway, which is manifested as the release of signaling molecule, phosphorylation of signal pathway protein, and activation of transcription factor. The continuous reaction can make the induction and response of copper ions reach a stable state that satisfies a system of differential equations

Due to (1) and (2), the relationship between pheromone concentration and copper ion concentration is solved by

where K2 = τ1/μ2 is the Mie constant of pheromone, Kbase = γ1/μ1 is the basal expression level of pheromone, and K1 = β1/μ1 is the Mie constant of pheromone mRNA. Thus, Eq. (14) can be transformed into

When combining (15) and (19) to solve (17c), the relationship between pheromone concentration and fluorescent protein expression level is given by

where Kleak = γ2/μ3 is the basal expression level of fluorescent protein mRAN and Kmapk = β2/μ3 is the maximum expression level of fluorescent protein. According to (17d) and (20), the transfer function of copper ion induced fluorescence protein expression is computed by

where Ky = βy/μy is the dissociation constant of fluorescent protein. Substituting (19) and (20) into (21), a transfer function is given as below

From (22), we define the transfer function of copper ion detection y=Φ(x) and its inverse function x=Φ-1(y). When performing online detection, the fluorescence intensity measured by flow cytometer is substituted into the inverse function to compute an estimated concentration of copper ions.

According to the experimental results of copper ion detection, we estimate the parameters of transfer function by using least squares estimation method. Table 1 gives parameter estimation.

Table 1 Parameter estimation

Data fitting in Fig. 3 shows that the model curve is consistent with the trend of experimental data. The corresponding evaluation index R2 is 0.94. This result indicates that the proposed model for copper ion detection can accurately description the signal reaction process from copper ion to fluorescence expression. Furthermore, Fig. 3 displays that our biosensor can approximate the linear amplification of fluorescence intensity, which has good stability and sensitivity. To analyze the effect of linear amplification, the signal amplification processes of copper ion metabolism and MAPK signal pathway are showed in Fig. 4(a) and Fig. 4(b), respectively. The more pheromones induced by the increased copper ions activate the protein kinase network to generate more fluorescent proteins. Such a positive growth signal cascade can enhance the detection accuracy of copper ion biosensor.

Figure 3 Data fitting for copper ion detection.

Figure 4 Signal amplification curves of copper ion metabolism and MAPK signal pathway.

The copper ion metabolism process is influenced by four parameters, i.e., K1, Kbase, K2, and Kx. Fig. 5 gives the growth curves of pheromone induced by copper ion under different levels of four parameters. It is clear to see that the lager K1 and K2 are, the more sensitive the pheromone is to the changes of copper ion concentration. However, the sensitivity decreases with the increase of Kx. Different Kbase affects the response interval of pheromone, but little affects sensitivity.

Figure 5 Growth curves of pheromone induced by copper ion under different levels of four parameters.

The MAPK signal pathway is influenced by four parameters, i.e., Ky, Kleak, Kmapk, and K2. Fig. 6 gives the growth curves of fluorescent protein induced by pheromone under different levels of four parameters. We can attain the similar results to Fig. 5. The increased Kmapk and K2 can help to improve the response ability of fluorescent protein to pheromone.

Figure 6 Growth curves of fluorescent protein induced by pheromone under different levels of four parameters.

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