Gene pathway model
Preface
1.1 Why do we need a model
Developing a mathematical model of the systems we established (or plan to establish) is a vital step in our project. A model, namely, is a mathematical representation of the system. The model could help us in the following aspects:
- Confirming that our system has the potential to work properly. For example, as for the Test System, the model could present us whether both the self-activating component and safety-catch component function or not.
- Determining optimal conditions for the system.
- Estimating biological parameters.
- Understanding the mechanism of the system better.
- Optimizing the design of our final project.
1.2 What can we get from modeling
Based on logistic function, Monod kinetics,etc, we can describe a cell or the group of cells mathematically. By quantifying cells, a number of parameters can be predicted, including the colony growth rate of Pichia pastoris , the expressions of proteins and the changes of the PH during the growth of cells. More significantly, the model of the project gives us a visualize and intuitive result, which could guide the experiment before running it.
1.3 General assumptions
In modeling the growth of the pastoris, in order to analyze the growth of cells efficiently and mathematically, we use unsegregated model, in which average cellular properties and balanced growth are assumed. Also, each cell is deemed identical, meaning that population is regarded as one-component solute and cells are produced of the same quality with the common components — at various stage of the co-culture, new cells are assumed to be equal to old cells.
The model is also unstructured, in the sense that the intermediate reactions are omitted.To simply the model, we assume that the environment remain unchanged during the growth of Pichia pastoris and the toxic effects of the metabolites from these reactions are ignored, shows the K is a non-time-value parameter. Likewise, the popluation of Pichia pastoris should not exceed the carrying capacity of the bioreactor.
And in modeling the gene path, we ignoring the fluctuation and the external noise in expression process, so that the expression speed of the protein can be regard as a Time-dependent function and later be described as a partial differential equation.
Developing a mathematical model of the systems we established (or plan to establish) is a vital step in our project. A model, namely, is a mathematical representation of the system. The model could help us in the following aspects:
- Confirming that our system has the potential to work properly. For example, as for the Test System, the model could present us whether both the self-activating component and safety-catch component function or not.
- Determining optimal conditions for the system.
- Estimating biological parameters.
- Understanding the mechanism of the system better.
- Optimizing the design of our final project.
1.2 What can we get from modeling
Based on logistic function, Monod kinetics,etc, we can describe a cell or the group of cells mathematically. By quantifying cells, a number of parameters can be predicted, including the colony growth rate of Pichia pastoris , the expressions of proteins and the changes of the PH during the growth of cells. More significantly, the model of the project gives us a visualize and intuitive result, which could guide the experiment before running it.
1.3 General assumptions
In modeling the growth of the pastoris, in order to analyze the growth of cells efficiently and mathematically, we use unsegregated model, in which average cellular properties and balanced growth are assumed. Also, each cell is deemed identical, meaning that population is regarded as one-component solute and cells are produced of the same quality with the common components — at various stage of the co-culture, new cells are assumed to be equal to old cells.
The model is also unstructured, in the sense that the intermediate reactions are omitted.To simply the model, we assume that the environment remain unchanged during the growth of Pichia pastoris and the toxic effects of the metabolites from these reactions are ignored, shows the K is a non-time-value parameter. Likewise, the popluation of Pichia pastoris should not exceed the carrying capacity of the bioreactor.
And in modeling the gene path, we ignoring the fluctuation and the external noise in expression process, so that the expression speed of the protein can be regard as a Time-dependent function and later be described as a partial differential equation.
Modeling
Modeling of colony growth of Pichia pastoris
1. Using logistic function to predict the growth of Pichia pastoris
By the hypothesis above, we can assume that Pichia pastoris and yeast follow logistic growth.
In the equation, the concentration of Pichia pastoris should not exceed the carrying capacity of the bioreactor, K, which is assumed to be non-time-varying. μP-S is specific growth rate.dP represents the specific cell death rate of Pichia pastoris.
2. Deriving specific growth rate based on Monod kinetics
Specific growth rate, μ, affected by the limited substrate and the fluctuation of the pH, is expressed using Monod kinetics.
a. Specific growth rate with the concentration of substrate
By assuming the total substrate uptake follows Michaelis-Menten kinetics and the substrate is used for non-growth associated metabolic maintenance during stationary phase and microbial growth, the substrate uptake rate rs is expressed as:
In which KmaxP-S is the maximumsubstrate uptake rate by cells. S represents substrate concentration, P represents the concentration of Pichia pastoris,YP-S is the yield coefficient of the biomass to the substrate consumed, and KP-S is the substrate affinity. With constant maintenance requirement assumed, m could be regarded as a maintenance constant.
Therefore, the specific growth rate can be expressed as:
And assuming S >> KP-S the maximum growth rate should be:
And the substrate affinity can be expressed as:
Thus, the specific growth rate can be reformulated into:
The substrate consumption rate becomes:
b.The influence of the fluctuation of the pH
The pH of a culture may influence both the maximum growth rate and the inhibitory potentials of substances, particularly those of organic acids. To account for the effect of pH on μP-Smax the following expression is used:
where H+ is the hydrogen ion concentration；μP-Smax, KH, and KOH are constants.
3. The concentration of substrate
In order to keep the system working properly, we should keep adding substrates. In our system, we use formaldehyde as the carbon source of Pichia pastoris. For the sake of simple treatment, we assume that the addition of the substrate changes with time is a constant. That is:
kS-add is a constant.
Therefore, the concentration of the substrate can be expressed as
Combined with the equations above, the differential equation of substrate concentration according to time is:
By the hypothesis above, we can assume that Pichia pastoris and yeast follow logistic growth.
In the equation, the concentration of Pichia pastoris should not exceed the carrying capacity of the bioreactor, K, which is assumed to be non-time-varying. μP-S is specific growth rate.dP represents the specific cell death rate of Pichia pastoris.
2. Deriving specific growth rate based on Monod kinetics
Specific growth rate, μ, affected by the limited substrate and the fluctuation of the pH, is expressed using Monod kinetics.
a. Specific growth rate with the concentration of substrate
By assuming the total substrate uptake follows Michaelis-Menten kinetics and the substrate is used for non-growth associated metabolic maintenance during stationary phase and microbial growth, the substrate uptake rate rs is expressed as:
In which KmaxP-S is the maximumsubstrate uptake rate by cells. S represents substrate concentration, P represents the concentration of Pichia pastoris,YP-S is the yield coefficient of the biomass to the substrate consumed, and KP-S is the substrate affinity. With constant maintenance requirement assumed, m could be regarded as a maintenance constant.
Therefore, the specific growth rate can be expressed as:
And assuming S >> KP-S the maximum growth rate should be:
And the substrate affinity can be expressed as:
Thus, the specific growth rate can be reformulated into:
The substrate consumption rate becomes:
b.The influence of the fluctuation of the pH
The pH of a culture may influence both the maximum growth rate and the inhibitory potentials of substances, particularly those of organic acids. To account for the effect of pH on μP-Smax the following expression is used:
where H+ is the hydrogen ion concentration；μP-Smax, KH, and KOH are constants.
3. The concentration of substrate
In order to keep the system working properly, we should keep adding substrates. In our system, we use formaldehyde as the carbon source of Pichia pastoris. For the sake of simple treatment, we assume that the addition of the substrate changes with time is a constant. That is:
kS-add is a constant.
Therefore, the concentration of the substrate can be expressed as
Combined with the equations above, the differential equation of substrate concentration according to time is:
Modeling of single Cell Gene Pathway
In Pichia pastoris, three heterologous genes are expressed：PelA, SLAC and VP. First we model their expressions separately. In addition, the expression of alkaline protein in Pichia pastoris is beneficial to regulate the low pH of the system and maintain the stability of the system.
During the growth of Pichia pastoris, the number of Pichia pastoris can be calculated by the following equation:
Where NumP is the number of Pichia pastoris, P is the concentration of Pichia pastori, V is the volume of the bioreactor, ρis the mass of a Pichia pastori.
1. Kinetic Model of PelA Gene expression
The kinetic equation of PelA gene transcription is：
where trcPelA is the transcription rate of PelA，degPelA-mRNA is the degradation rate of mRNAPelA.
The last part of this equation takes into account the effect of pH on gene expression. KH-PelA , KOH-PelA and nPelA are constants.In view of the fact that the expression of different genes varies from the change of pH, add parameters(ngene) to adjust the equation.
The kinetic equation of PelA gene transcription is：
2. Kinetic Model of SLAC Gene expression
The kinetic equation of SLAC gene transcription is：
where trcSLAC is transcription rate of SLAC，degSLAC-mRNA is the degradation rate of mRNASLAC. As discussed above, KH-PelA , KOH-SLAC and nSLAC are constants .
The kinetic equation of SLAC gene transcription is：
where trlSLAC is translation rate of mRNASLAC，degSLAC-Pro is the degradation rate of ProSLAC.
3. Kinetic Model of VP Gene expression
The kinetic equation of VP gene transcription is：
where trcVP is transcription rate of VP，degVP-mRNA is the degradation rate of mRNAVP. As discussed above, KH-VP , KOH-VP and nVP are constants .
The kinetic equation of VP gene transcription is：
4. Kinetic Model of alkaline protein Gene expression
The kinetic equation of alkaline protein(abbreviated as a-Pro) gene transcription is：
where trca-Pro is transcription rate of a-Pro，dega-Pro-mRNA is the degradation rate of mRNAa-Pro. As discussed above, KH-a-Pro , KOH-a-Pro and na-Pro are constants .
The kinetic equation of a-Pro gene transcription is：
where trla-Pro is translation rate of mRNAa-Pro，dega-Pro is the degradation rate of Proa-Pro.
During the growth of Pichia pastoris, the number of Pichia pastoris can be calculated by the following equation:
Where NumP is the number of Pichia pastoris, P is the concentration of Pichia pastori, V is the volume of the bioreactor, ρis the mass of a Pichia pastori.
1. Kinetic Model of PelA Gene expression
The kinetic equation of PelA gene transcription is：
where trcPelA is the transcription rate of PelA，degPelA-mRNA is the degradation rate of mRNAPelA.
The last part of this equation takes into account the effect of pH on gene expression. KH-PelA , KOH-PelA and nPelA are constants.In view of the fact that the expression of different genes varies from the change of pH, add parameters(ngene) to adjust the equation.
The kinetic equation of PelA gene transcription is：
2. Kinetic Model of SLAC Gene expression
The kinetic equation of SLAC gene transcription is：
where trcSLAC is transcription rate of SLAC，degSLAC-mRNA is the degradation rate of mRNASLAC. As discussed above, KH-PelA , KOH-SLAC and nSLAC are constants .
The kinetic equation of SLAC gene transcription is：
where trlSLAC is translation rate of mRNASLAC，degSLAC-Pro is the degradation rate of ProSLAC.
3. Kinetic Model of VP Gene expression
The kinetic equation of VP gene transcription is：
where trcVP is transcription rate of VP，degVP-mRNA is the degradation rate of mRNAVP. As discussed above, KH-VP , KOH-VP and nVP are constants .
The kinetic equation of VP gene transcription is：
4. Kinetic Model of alkaline protein Gene expression
The kinetic equation of alkaline protein(abbreviated as a-Pro) gene transcription is：
where trca-Pro is transcription rate of a-Pro，dega-Pro-mRNA is the degradation rate of mRNAa-Pro. As discussed above, KH-a-Pro , KOH-a-Pro and na-Pro are constants .
The kinetic equation of a-Pro gene transcription is：
where trla-Pro is translation rate of mRNAa-Pro，dega-Pro is the degradation rate of Proa-Pro.
Modeling of the degradation of Banana components
The components of banana stem include cellulose, hemicellulose, lignin and pectin.The purpose of our project is to use the three enzymes heterogeneously expressed in Pichia pastoris to degrade lignin, pectin and hemicellulose to obtain higher quality fiber.
Simple Mass Action doesn't fit enzymatic reactions well. The reason for it is the fact that the enzyme and the substrate form a complex, which is then converted to a product and the original enzyme.In the process of substrate degradation, the degradation rate of enzyme satisfies Michaelis-Menten equation：
It has several parameters including the maximal reaction rate vmax , the Michaelis constant Km and the turnover number kcat.
1. Kinetic Model of degradation of pectin and galacturonic acid
In the process of pectin degradation, the vast majority of the degradation product is galacturonic acid(abbreviated as GA), which lead to the decrease of pH. The equation of GA is as follows:
2. Kinetic Model of degradation of lignin
3. Kinetic Model of degradation of hemicellulose
Simple Mass Action doesn't fit enzymatic reactions well. The reason for it is the fact that the enzyme and the substrate form a complex, which is then converted to a product and the original enzyme.In the process of substrate degradation, the degradation rate of enzyme satisfies Michaelis-Menten equation：
It has several parameters including the maximal reaction rate vmax , the Michaelis constant Km and the turnover number kcat.
1. Kinetic Model of degradation of pectin and galacturonic acid
2. Kinetic Model of degradation of lignin
3. Kinetic Model of degradation of hemicellulose
Modeling of the degradation of Banana components
In our project, pH, as an important environmental variable, affects many processes, such as gene expression, enzyme activity, yeast growth, and so on.
1. Effect of pH on enzyme activity
The effect of acid H+ ions or basic OH− ions on the activity of an enzyme is probably caused by a change in stereo configuration at or in the neighborhood of the active sites (Fersht, 1984; Whitaker, 1994). As in almost all protonation reactions, these reactions occur very rapidly. The different configurations are instantaneously in equilibrium. The protonated and hydroxylated enzymes are assumed to be completely inactive or at least less active. This can be represented by the following mechanisms:
where KEH and KEOH are the equilibrium constants of the reactions.
The water dissociation is defined, as usual, as:
The amount of EnH and EnOH can now be expressed in terms of actual amount of active enzyme and pH by:
The total amount of enzyme in any configuration has to remain constant:
` According to the equation above, solving for En, an expression is obtained for the active enzyme at any H+ concentration (or pH):
Again, this model, for the amount of available active enzyme configuration, can be converted into an apparent activity (Act), as for the temperature model. This results in:
Again, in this equation, k s and only appear in combination with each other. It is therefore impossible to estimate both variables at the same time. Both parameters are therefore combined in a new parameter called Act 0 . This results in the final equation:
The equation describes how the activity of the total pool of enzyme changes with H + concentration over the entire range from pH0 up to pH14.
2. Effect of pH on the growth of Pichia pastoris
The pH effect on bacterial growth may be directly attributed to the H+ in the medium. The hydrogen ion has been described as a noncompetitive inhibitor to the growth of some microorganisms at acidic pH. The hydrogen ions may inhibit one metabolic reaction that is the rate-limiting step and thus limit the growth rate of the culture. In general, the effect of H+ on the specific growth rate may be represented as follows:
where H+ is the hydrogen ion concentration；μP-Smax*, KH, and KOH are constants.
3. Effect of pH on gene expression
The kinetic equation of gene transcription is：
where trcgene is transcription rate of gene，deggene-mRNA is the degradation rate of mRNAgene.
The last part of this equation takes into account the effect of pH on gene expression. KH-gene , KOH-gene and ngene are constants.
In view of the fact that the expression of different genes varies with the change of pH, add parameters(ngene) to adjust the equation.
1. Effect of pH on enzyme activity
The effect of acid H+ ions or basic OH− ions on the activity of an enzyme is probably caused by a change in stereo configuration at or in the neighborhood of the active sites (Fersht, 1984; Whitaker, 1994). As in almost all protonation reactions, these reactions occur very rapidly. The different configurations are instantaneously in equilibrium. The protonated and hydroxylated enzymes are assumed to be completely inactive or at least less active. This can be represented by the following mechanisms:
where KEH and KEOH are the equilibrium constants of the reactions.
The water dissociation is defined, as usual, as:
The amount of EnH and EnOH can now be expressed in terms of actual amount of active enzyme and pH by:
The total amount of enzyme in any configuration has to remain constant:
` According to the equation above, solving for En, an expression is obtained for the active enzyme at any H+ concentration (or pH):
Again, this model, for the amount of available active enzyme configuration, can be converted into an apparent activity (Act), as for the temperature model. This results in:
Again, in this equation, k s and only appear in combination with each other. It is therefore impossible to estimate both variables at the same time. Both parameters are therefore combined in a new parameter called Act 0 . This results in the final equation:
The equation describes how the activity of the total pool of enzyme changes with H + concentration over the entire range from pH0 up to pH14.
2. Effect of pH on the growth of Pichia pastoris
The pH effect on bacterial growth may be directly attributed to the H+ in the medium. The hydrogen ion has been described as a noncompetitive inhibitor to the growth of some microorganisms at acidic pH. The hydrogen ions may inhibit one metabolic reaction that is the rate-limiting step and thus limit the growth rate of the culture. In general, the effect of H+ on the specific growth rate may be represented as follows:
where H+ is the hydrogen ion concentration；μP-Smax*, KH, and KOH are constants.
3. Effect of pH on gene expression
The kinetic equation of gene transcription is：
where trcgene is transcription rate of gene，deggene-mRNA is the degradation rate of mRNAgene.
The last part of this equation takes into account the effect of pH on gene expression. KH-gene , KOH-gene and ngene are constants.
In view of the fact that the expression of different genes varies with the change of pH, add parameters(ngene) to adjust the equation.
Modeling of the degradation of Banana components
In this system, there are several reactions that are directly related to the concentration of H+.They are:
Therefore, the differential equation can be written as :
Therefore, the differential equation can be written as :
3-D Structural Model
1.Why do we construct the protein 3-D structures
2. I-TASSER
3.The structures of the three enzymes with different positions of His-tag
4.Analysis the possibility level of the chelation mathematical 
5.Conclusion
6.References1.J Yang, Y Zhang. I-TASSER server: new development for protein structure and function predictions, Nucleic Acids Research, 43: W174-W181, 2015.
2.C Zhang, PL Freddolino, Y Zhang. COFACTOR: improved protein function prediction by combining structure, sequence and protein–protein interaction information. Nucleic Acids Research, 45: W291-W299, 2017.
3. Jmol: an open-source Java viewer for chemical structures in 3D.
