Model | UM

I. Introduction

Under continuous illumination, all intermediates enter a plateau, the excited XeR is mainly composed of the intermediate molecules in the rate determination transition, which controls most of the photo-induced current. In a previous experiment, the protein XeR was heterologously expressed in the E. coli C41 strain and showed that pH changes upon illumination in liposome suspension measured under different pH values. It is supposed that the upstream follow the change as well, which means the photo-induced current is determined by external $H^{+}$ concentration. Then, firstly paying attention to the intermediates in the steady photo-induced current, we simplify the XeR photocycle and build a model of photo-induced current due to the $H^{+}$ transport via XeR.

II. Definition of Variables

The extracellular concentration of $H^{+} : C_{H^{+^{ec}}}$
The cytoplasmic concentration of $H^{+} : C_{H^{+^{c p}}}$
The light intensity: $L$
The photoinduced current: $I$
The total concentration of excited XeR involved in the photo-induced current cycle: $P$
The total amount of the pXeR protein expressed in the plasma membrane: $P_{t o t a l}$
The ratio of $P$ to the unexcited original pXeR: $r$
The concentration of XeR∗ converted from original XeR in unit time in the photoreaction process: $C_{X e R \to X e R *}$
The E. coli DH5 alpha density: $ρ_{DH 5}^{0}$
The volume of E. coli DH5 alpha growth medium: $V_{DH 5}$
The volume of a single E. coli DH5 alpha: $V_{DH 5}^{s in g l e}$
The volume of target seawater: $V_{se a}$
The concentration of initial extracellular XeR: $C_{X e R} ∣_{ini t}$
The concentration of initial extracellular $H^{+}$ : $C_{H^{+}}^{ec} ∣_{ini t}$
The concentration of $H^{+}$ free intermediate: $C_{f ree}$
The concentration of $H^{+}$ bound intermediate: $C_{b o u n d}$
The concentration of original XeR: $C_{X e R}$
The concentration of excited XeR: $C_{X e R} *$
The chemical reaction constant of the fast transition process: $k_{f a s t}$
The chemical reaction constant of the rate determining process: $k_{r a t e}$
The association and dissociation rate constants about $H^{+}$ binding to the extracellular site in XeR: $k_{a}, k_{d}$
The proportionality of XeR to XeR * process: $k_{p h o t o}$
The chemical reaction constant of the reverse fast transition process: $k_{f a s t}^{re v}$

The ratio of excited XeR to unexcited XeR: $k$

III. Modelling

Because of its intrinsic properties as a proton pump, XeR is completely independent of the ion conditions. Moreover, the photocycle of XeR contains a microsecond-duration phase, which is usually assigned to the multistep reaction of the release of $H^{+}$ and a millisecond-duration phase, consisting of relaxation and reuptake of the ion. Therefore, the whole process is divided into two parts. One is the fast transition of intermediates corresponding to $H^{+}$ relaxation and reuptake processes, and the other one is the rate-determining transition corresponding to $H^{+}$ binding processes.

Then we could obtain the following equations with the law of mass action in chemical reaction theory:

\frac{d C _{X e R}}{d t} = k_{r a t e} C_{b o u n d} - C_{X e R \to X e R^{*}} (1)

\frac{d C _{X e R^{*}}}{d t} = C_{X e R \to X e R^{*}} + k_{f a s t}^{in v} C_{H^{+}}^{c p} C_{f ree} - k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} (2)

\frac{d C _{f ree}}{d t} = k_{d} C_{b o u n d} + k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} - k_{a} C_{H^{+}}^{ec} C_{f ree} - k_{f a s t}^{in v} C_{H^{+}}^{c p} C_{f ree} (3)

\frac{d C _{b o u n d}}{d t} = k_{a} C_{H^{+}}^{ec} C_{f ree} - (k_{d} + k_{r a t e}) C_{b o u n d} (4)

Alternatively, the equation related to the total concentration of excited XeR molecules involved in the photoinduced current cycle holds as follows:

P = C_{X e R^{*}} + C_{f ree} + C_{b o u n d} (5)

Under continuous illumination, all intermediates enter a plateau and all mass-balance equations above equal to zero. Combining all equations and substituting the value into the above equation Eq. 4, we can get the following equation:

P = k_{r a t e} C_{b o u n d} (\frac{1}{k _{f a s t}} + \frac{\frac{k _{d} + k _{r a t e}}{k _{a}}}{k _{r a t e} [ H ^{+} ]} + \frac{1}{k _{r a t e}}) (6)

All transition rates are equal. Thus, we can express the photo-induced current at a steady state as the multiplicity of $C_{b o u n d}$ , $k_{r a t e}$ and Faraday’s constant, F, as follows:

I = k_{r a t e} F C_{b o u n d} = \frac{FP}{\frac{1}{k _{f a s t}} \frac{\frac{k _{d} + k _{r a t e}}{k _{a}}}{k _{r a t e [H^{+}]}} + \frac{1}{k _{r a t e}}} (7)

Considering that other rate constants ( $k_{f a s t}, k_{a}, an d k_{d}$ ) are further larger than the $k_{r a t e}$ value, the above equation is reduced to:

I = k_{r a t e} F C_{b o u n d} = \frac{k _{r a t e} PF [ H ^{+} ]}{\frac{k _{d}}{k _{a}} + [ H ^{+} ]} (8)

The light-induced currents are defined by kinetic parameters $I_{M A X}$ which represents the maximal photoinduced current and $K_{0.5}$ which represents the ion concentration necessary for the induction of half-maximal current. Using the data from whole-cell patch-clamp experiments, where NsXeR was expressed in NG108-15 and iterative nonlinear least-squares methods, we can have a Michaelis-Menten-fitted equation to describe a single saturable component of photoinduced current (I):

I = \frac{I _{M A X} [ s ]}{K _{0.5} + [ s ]} (9)

where [s] means the concentration of the transportable anion in the perfusion buffer.

Then substitute the multiplicity $k_{r a t e}$ PF for $I_{M A X}$ and the ratio $\frac{k _{d}}{k _{a}}$ for $K_{0.5}$ , the Michaelis-Menten-type equation turns to:

I = k_{r a t e} F C_{b o u n d} = \frac{I _{M A X} [ H ^{+} ]}{K _{0.5} + [ H ^{+} ]} (10)

With the ratio r, express P express as:

P = \frac{r}{r + 1} P_{t o t a l} (11)

After contributing the model in the stable situation, we now consider the $H^{+}$ concentration variation in the non-equilibrium situation to further simulate the whole complex system:

\frac{d C _{H^{+}}^{ec}}{d t} = k_{d} C_{b o u n d} - C_{f ree} C_{H^{+}}^{ec} k_{a} + k_{f a s t}^{re v} C_{f ree} C_{H^{+}}^{c p} - k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} (12)

To better proceed, we add one assumption as follows:

Photoreaction rate is proportional to the concentration of XeR, which means $C_{X e R \to X e R^{*}} \propto C_{X e R}$ .

Combined with the above assumptions, $C_{X e R \to X e R^{*}}$ can be written as:

C_{X e R \to X e R^{*}} = k_{p h o t o} C_{X e R} (13)

With $Eq . 1, 2, 3, 4, 13 and 14$ , we can get the final equation system:

⎩ ⎨ ⎧ \frac{d C _{H^{+}}^{ec}}{d t} = k_{d} C_{b o u n d} - C_{f ree} C_{H^{+}}^{ec} k_{a} + k_{f a s t}^{re v} C_{f ree} C_{H^{+}}^{c p} - k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} \frac{d C _{f ree}}{d t} = k_{d} C_{b o u n d} + k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} - k_{a} C_{H^{+}}^{ec} C_{f ree} - k_{f a s t}^{in v} C_{H^{+}}^{c p} C_{f ree} \frac{d C _{b o u n d}}{d t} = k_{a} C_{H^{+}}^{ec} C_{f ree} - (k_{d} + k_{r a t e}) C_{b o u n d} \frac{d C _{X e R}}{d t} = k_{r a t e} C_{b o u n d} - k_{p h o t o} C_{X e R} \frac{d C _{X e R^{*}}}{d t} = k_{p h o t o} C_{X e R} + k_{f a s t}^{re v} C_{H^{+}}^{c p} C_{f ree} - k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} (14)

In the meanwhile, we have the conservation law of $H^{+}$ :

C_{H^{+}}^{ec} V_{se a} + C_{H^{+}}^{c p} V_{DH 5}^{s in g l e} ρ_{DH 5}^{0} V_{DH 5} = C_{H^{+}}^{ec} ∣_{ini t} V_{se a} (15)

And as $V_{se a}$ is actually $V_{DH 5}$ in our experiment, the above equation can be written as:

C_{H^{+}}^{c p} = \frac{( C _{H^{+}}^{ec} ∣ _{ini t} - C _{H^{+}}^{ec} )}{V _{DH 5}^{s in g l e} ρ _{DH 5}^{0}} (16)

Then substitute Eq.17 into Eq.15, we can obtain:

⎩ ⎨ ⎧ \frac{d C _{H^{+}}^{ec}}{d t} = k_{d} C_{b o u n d} - C_{f ree} C_{H^{+}}^{ec} k_{a} + k_{f a s t}^{re v} C_{f ree} \frac{( C _{H^{+}}^{ec} ∣ _{ini t} - C _{H^{+}}^{ec} )}{V _{DH 5}^{s in g l e} ρ _{DH 5}^{0}} - k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} \frac{d C _{f ree}}{d t} = k_{d} C_{b o u n d} + k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} - k_{a} C_{H^{+}}^{ec} C_{f ree} - k_{f a s t}^{in v} C_{f ree} \frac{( C _{H^{+}}^{ec} ∣ _{ini t} - C _{H^{+}}^{ec} )}{V _{DH 5}^{s in g l e} ρ _{DH 5}^{0}} \frac{d C _{b o u n d}}{d t} = k_{a} C_{H^{+}}^{ec} C_{f ree} - (k_{d} + k_{r a t e}) C_{b o u n d} \frac{d C _{X e R}}{d t} = k_{r a t e} C_{b o u n d} - k_{p h o t o} C_{X e R} \frac{d C _{X e R^{*}}}{d t} = k_{p h o t o} C_{X e R} + k_{f a s t}^{re v} C_{f ree} \frac{( C _{H^{+}}^{ec} ∣ _{ini t} - C _{H^{+}}^{ec} )}{V _{DH 5}^{s in g l e} ρ _{DH 5}^{0}} - k_{f a s t} C_{X e R^{*}} C_{H^{+}}^{ec} (17)

Notably, the initial conditions are set as:

⎩ ⎨ ⎧ C_{H^{+}}^{ec} = C_{H^{+}}^{ec} ∣_{ini t} C_{f ree} = 0 C_{b o u n d} = 0 C_{X e R} = C_{X e R} ∣_{ini t} C_{X e R^{*}} = 0 (18)

IV. Simulation

This model’s simulation is visualized enough to show the real-time variation of the concentration of hydrogen, XeR and excited XeR in the medium, allowing us to calculate the efficiency of our system.

Notably, the initial XeR concentration is $1.0 \times 1 0^{5}$ times of our actual emxperimental data for better demostration.

Figure 2(a). real-time concentration of hygrogen variation; figure 2(b). real-time concentration of unexcited XeR; figure 2(c). real-time concentration of excited XeR^*.

V. Conclusion

We can observe that our model works well for the simulation shows successful pH increase from 5.5 to 7. However due to the strong assumptions, we cannot use the model to precisely predict our entire system. Take the XeR concentration for an instance, the actual XeR concentration maybe much lower than what we use during modelling, which will definitely influence the final result.

Despite our experiment is more complicated than the condition the model simulates, it helps a lot in directing us to modify the rates with regulate XeR proteins, then become in line with the parameters of the model. And the model also allows us to know that our assumptions and designation are feasible before furthermore detailed complement.

In a nutshell, the imperfect model provides us the way to adjust our experiment and data from experiment can optimize the model in turn, and other teams who want to use such proteins to control ocean acidification can be inspired by our model as well.

VI. Reference

Seki, A., Miyauchi, S., Hayashi, S., Kikukawa, T., Kubo, M., Demura, M., Ganapathy, V., & Kamo, N. (2007). Heterologous expression of Pharaonis Halorhodopsin in xenopus laevis oocytes and electrophysiological characterization of its light-driven Cl− pump activity. Biophysical Journal, 92(7), 2559-2569. https://doi.org/10.1529/biophysj.106.093153
Shevchenko, V., Mager, T., Kovalev, K., Polovinkin, V., Alekseev, A., Juettner, J., Chizhov, I., Bamann, C., Vavourakis, C., Ghai, R., Gushchin, I., Borshchevskiy, V., Rogachev, A., Melnikov, I., Popov, A., Balandin, T., Rodriguez-Valera, F., Manstein, D. J., Bueldt, G., … Gordeliy, V. (2017). Inward H ⁺ pump xenorhodopsin: Mechanism and alternative optogenetic approach. Science Advances, 3(9). https://doi.org/10.1126/sciadv.1603187
Team:sjtu-biox-Shanghai/Modeling. (n.d.). 2015.igem.org. https://2015.igem.org/Team:SJTU-BioX-Shanghai/Modeling

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