Difference between revisions of "Team:OUC-China/Model"

Latest revision as of 14:00, 18 November 2017

Model

Overview

There were two different types of models built to help our team gain the better understanding and further implementation of our project.

Firstly, we built pathway models (xylose pathway and cellobiose pathway) in MATLAB using Simbiology Toolbox by a set of enzymatic reaction kinetic equations in order to confirm our pathway design of producing ethanol. And we did sensitivity analyses using local methods to explore the rate-determined step in our pathways, which would further help us optimize the pathways for higher production.

Secondly, we built agent-based models (ABM, a wonderful type of models to investigate complex dynamic systems) using Netlogo. We utilized these models to run real-time simulations of adhesion platform we built, as well as explore the properties of the system. This allowed to show us vividly how would the system behave (e.g. coculture growth, binding ratios over time), and to prove our design.

Our models are modefied by our wet lab date and gave a wonderful guidance to our experiment in return.

Supplementary and Coding could be download here

Xylose Pathway

Overview

The fundamental parts of our project is two pathways in S.cerevisiae, which allow the strain to ferment the xylose and cellobiose to ethanol separately. In xylose pathway model, we would describe the kinetic system in some mathematical equations, and explore the main factors influencing the production of ethanol by sensitivity analysis, which were then given to the wet lab for implementation and the better understanding of the system.

1. Describe the xylose pathway using ordinary differential equations(ODEs).
2. Explore the rate-determined step (RDS) of the xylose pathway using sensitivity analysis.

Assumptions

(I) All enzyme kinetics obey Michaelis-Menten functions.
(II) All coenzymes are finite and stable.
(III) S.cerevisiae grows following logistic equation.

Reactions

We can simplify our pathway reactions to be followings:

\[XYLe\leftrightarrow XYL\] \[XYL+NADPH\leftrightarrow NADP+XYT\] \[XYT+NAD\leftrightarrow NADH+XYLU\] \[XYLU+ATP\rightarrow [5-P]\] \[[5-P]\rightarrow xEthanol\] \[C_X\rightarrow \emptyset\] x=15(Estimated by experimental data)

Figure 1.1 Schema of xylose pathway reactions.

ODEs

To simulate the consumption of xylose and production of ethanol, we use ordinary differential equations to model the reactions above. And ODEs are given as follows:

\[\frac{d(XYLe)}{dt}=\left(-V_{trsp,in}+V_{trsp,out}\right)C_X\] \[\frac{d(XYL)}{dt}=\left(V_{trsp,in}-V_{trsp,out}\right)C_X-V_{XR}\] \[\frac{d(XYT)}{dt}=V_{XR}-V_{XDH}\] \[\frac{d(XYLU)}{dt}=V_{XDH}-V_{XK}\] \[\frac{d[5-P]}{dt}=V_{XK}-V_{pro}\] \[\frac{d(Ethanol)}{dt}=V_{pro}\] \[\frac{dC_X}{dt}=V_{grwmax}\left(1-\frac{C_X}{C_{XS}}\right)C_X\] \[V_{trsp,in}=\frac{ V^{trsp}_{max}\frac{XYLe}{K^{trsp}_m}}{1+\frac{XYLe}{K^{trsp}_m}+\frac{a\frac{XYLe}{K^{trsp}_m}+1}{a\frac{XYL}{K^{trsp}_m}+1}\left(1+\frac{XYL}{K^{trsp}_m}\right)}\] \[V_{trsp,out}=\frac{ V^{trsp}_{max}\frac{XYL}{K^{trsp}_m}}{1+\frac{XYL}{K^{trsp}_m}+\frac{a\frac{XYL}{K^{trsp}_m}+1}{a\frac{XYLe}{K^{trsp}_m}+1}\left(1+\frac{XYLe}{K^{trsp}_m}\right)}\] \[V_{XR}=\frac{\frac{V^{XR}_{max}}{K^{NADPH}_i\cdot K\frac{XYL}{m}} \left(NADPH\cdot XYL-\frac{XYT\cdot NADP}{K_{eqXR}}\right)}{D1+D2}\] \[D1=1+\frac{K^{NADPH}_m\cdot XYL}{K^{NADPH}_i\cdot K^{XYL}_m}+\frac{K^{NADP}_m\cdot XYT}{K^{NADP}_i\cdot K^{XYT}_m}\] \[D2=\frac{NADPH}{K^{NADPH}_i}+\frac{NADP}{K^{NADP}_i}+\frac{NADPH\cdot XYL}{K^{NADPH}_i\cdot K^{XYL}_m}+\frac{K^{NADP}_m\cdot NADPH\cdot XYT}{K^{NADPH}_i\cdot K^{XYT}_m\cdot K^{NADP}_i}+\frac{K^{NADP}_m\cdot NADP\cdot XYL}{K^{NADPH}_i\cdot K^{XYL}_m\cdot K^{NADP}_i}+\frac{NADP\cdot XYT}{K^{NADP}_i\cdot K^{XYT}_m}+\frac{NADPH\cdot XYL\cdot XYT}{K^{NADPH}_i\cdot K^{XYL}_m\cdot K^{XYT}_i}+\frac{NADP\cdot XYL\cdot XYT}{K^{NADP}_i\cdot K^{XYT}_m\cdot K^{XYL}_i}\] \[V_{XDH}=\frac{\frac{V^{XDH}_{max}}{K^{NAD}_i\cdot K^{XYT}_m}\left(NAD\cdot XYT-\frac{XYLU\cdot NADH}{K_{eqXDH}}\right)}{D1+D2}\] \[D1=1+\frac{K^{NAD}_m\cdot XYT}{K^{NAD}_i\cdot K^{XYT}_m}+\frac{K^{NADH}_m\cdot XYLU}{K^{NADH}_i\cdot K^{XYLU}_m}\] \[D2=\frac{NAD}{K^{NAD}_i}+\frac{NADH}{K^{NADH}_i}+\frac{NAD\cdot XYT}{K^{NAD}_i\cdot K^{XYT}_i}+\frac{K^{NADH}_m\cdot NAD\cdot XYTLU}{K^{NAD}_i\cdot K^{XYLU}_m\cdot K^{NADH}_i}+\frac{K^{NAD}_m\cdot NADH\cdot XYT}{K^{NAD}_i\cdot K^{XYT}_m\cdot K^{NADH}_i}+\frac{NADH\cdot XYLU}{K^{NADH}_i\cdot K^{XYLU}_m}+\frac{NAD\cdot XYLU\cdot XYT}{K^{NAD}_i\cdot K^{XYT}_m\cdot K^{XYLU}_i}+\frac{NADH\cdot XYLU\cdot XYT}{K^{NADH}_i\cdot K^{XYLU}_m\cdot K^{XYT}_i}\] \[V_{XK}=\frac{V^{XK}_{max}\frac{XYLI}{K^{XYLU}_m}\cdot \frac{ATP}{K^{ATP}_m}}{1+\frac{XYLI}{K^{XYLU}_m}+\frac{ATP}{K^{ATP}_m}+\frac{XYLI}{K^{XYLU}_m}\cdot \frac{ATP}{K^{ATP}_m}}\] \[V_{pro}=\frac{V_{me}[5-P]}{K_{me}+[5-P]}\]

Reactans, enzymes and paramenters

Reactants	Meaning
XYLe	Xylos extracellular
XYL	Xylose intracellular
XYT	Xylitol
XYLU	Xylulose
[5-P]	D-xylulose-5-P
Ethanol	Ethanol

Table 1.1 Reactants in xylose pathway.

Enzymes	Meaning
transport	Transporter
XR	Xylose reductase
XDH	Xylitol dehydrogenase
XK	Xylulokinase

Table 1.2 Enzymes of reactions in xylose pathway.

Parameters	Meaning	Value	Unite	Reference
\(C_{XS}\)	Bacteria carrying capacity	0.1803	OD600	Estimated
\(V_{grwmax}\)	Maximum growth rate of yeast	0.1136	h^-1	Estimated
\(K_{me}\)	Michaelis constant of ethanol production	11.0	mM	Estimated
\(V_{me}\)	Maximum production rate of ethanol	1.0	mM/h	Estimated
\(\alpha\)	Coefficient	0.00062744	-	Estimated
\(K^{trsp}_m\)	Michaelis constant of xylose transporter	170.90134929	mM	Estimated
\(V^{trsp}_{max}\)	Maximum transportation rate of transporter	9.60568891	mM/h	Estimated
\(K_{eqXDH}\)	Equilibrium constant of XDH	7.0e^-7	mM	(Eliasson et al., 2001)
\(K^{NAD}_i\)	Inhibitory constant of NAD	0.435	mM	(Eliasson et al., 2001)
\(K^{NADH}_i\)	Inhibitory constant of NADH	0.008	mM	(Eliasson et al., 2001)
\(K^{XYLU}_i\)	Inhibitory constant of xylulose	243.3	mM	(Eliasson et al., 2001)
\(K^{XYT2}_i\)	Inhibitory constant of xylitol	81.2	mM	(Eliasson et al., 2001)
\(V^{NAD}_m\)	Michaelis constant of NAD	0.18	mM	(Eliasson et al., 2001)
\(K^{NADH}_m\)	Michaelis constant of NADH	0.07	mM	(Eliasson et al., 2001)
\(K^{XYLU}_m\)	Michaelis constant of xylulose	9.6	mM	(Eliasson et al., 2001)
\(K^{XYT2}_m\)	Michaelis constant of xylitol	18.6	mM	(Eliasson et al., 2001)
\(V^{XDH}_{max}\)	Maximum reaction rate of XDH	3456.0	mM/h	(Eliasson et al., 2001)
\(K^{ATP}_m\)	Michaelis constant of ATP	1.55	mM	(Richard et al., 2000)
\(K^{XYLU1}_m\)	Michaelis constant of xylulose	0.31	mM	(Richard et al., 2000)
\(V^{XK}_{max}\)	Maximum transportation rate of transporter	14240.0	mM/h	(Richard et al., 2000)
\(K_{eqXR}\)	Equilibrium constant of XR	575.0	mM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(K^{NADP}_i\)	Inhibitory constant of NADP	0.069	mM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(K^{NADPH}_i\)	Inhibitory constant of NADPH	0.0066	μM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(K^{XYL}_i\)	Inhibitory constant of xylose inside the cell	5982.0	mM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(K^{XYT1}_i\)	Inhibitory constant of xylitol	461.0	mM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(K^{NADP}_m\)	Michaelis constant of NADP	0.00709	mM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(K^{NADPH}_m\)	Michaelis constant of NADPH	0.0032	mM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(K^{XYL}_m\)	Michaelis constant of xylose inside the cell	67.7	mM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(K^{XYT1}_m\)	Michaelis constant of xylitoal	2029.0	mM	(Eliasson et al., 2001; Rizzi et al., 1988)
\(V^{XR}_{max}\)	Maximum reaction rate of XR	5637.0	mM/h	(Eliasson et al., 2001; Rizzi et al., 1988)

Table 1.3 Kinetic parameters of reactions in xylose pathway.

Simulation

We simulated xylose pathway in 90 hours, using Simbiology Toolbox in MATLAB. The simulation file can be download in Supplementary and Coding.

Figure 1.2 Simulation of xylose pathway in 90 hours.

We then compared the experimental data to simulation results.

Figure 1.3 Comparison between experimental data and simulation results.

Sensitivity Analysis

Since sensitivity analysis is a powerful tool to explore how the output fluctuates when an input change in a small range, we applied sensitivity analysis to investigate the rate-determined step in xylose pathway. Here we used local method and full normalized mode to avoid dimension difference.

\[S_{p_i}=\frac{p_i}{o_j(t)}\frac{\partial o_j(t)}{\partial p_i}\]

where \(p_i\) represents the ith parameter, and \(o_j\) represents the jth output, and \(S_{p_i}\) represents the sensitivity to the ith parameter.

We calculated the sensitivity of parameters in reactions changing over time, as well as numerical integration of each sensitivity.

Figure 1.4 Sensitivity over time of kinetic parameters in xylose pathway (top 10 shown).

Figure 1.5 Numerical integration of sensitivities of kinetic parameters in 300h.

According to the sensitivity analysis, two of the most significant parameters in xylose pathway are \(V_{max}^{XK}\) and \(V_{max}^{trsp}\), which means the rate-determining steps (RDS) are the reaction rate of XK (xylulokinase) and the transportation property of transporter. The results are mirror to our wet lab for xylose pathway optimization.

Cellobiose pathway

Overview

The fundamental parts of our project is two pathways in S.cerevisiae, which allow the strain to ferment the xylose and cellobiose to ethanol separately. Here we use the same methods as that of xylose pathway, to model cellobiose pathway.

This part would mainly contains two aspects:
1. Describe the cellobiose pathway using ODEs.
2. Explore the rate-determined step (RDS) of the cellobiose pathway using sensitivity analysis.

Assumptions

(I) All enzyme kinetics obey Michaelis-Menten functions.
(II) All coenzyme are finite and stable.
(III) S.cerevisiae grows following logistic equation.
Note that the assumptions are the same as that of xylose pathway.

Reactions

Similar to xylose pathway, cellobiose pathway reactions can be simplified as follows.

\[cbo\leftrightarrow cbi\] \[cbi\rightarrow glucose\] \[glucose\rightarrow Ethanol\] \[C_X\rightarrow \emptyset\]

Figure 2.1 Schema of cellobiose pathway reactions.

ODEs

To simulate the consumption of cellobiose and production of ethanol, we use ordinary differential equations to model the reactions above. And ODEs are given as follows:

\[\frac{d(cbo)}{dt}=\left(-V_{trsp,in}+V_{trsp,out}\right)C_X\] \[\frac{d(cbi)}{dt}=\left(V_{trsp,in}-V_{trsp,out}\right)C_X-V_{gh1-1}\] \[\frac{d(gloucose)}{dt}=V_{gh1-1}-V_{glco}\] \[\frac{d(Ethanol)}{dt}=V_{glco}\] \[\frac{dC_X}{dt}=V_{grwmax}\left(1-\frac{C_X}{C_{XS}}\right)C_X\] \[V_{trsp,in}-V_{trsp,out}=\frac{V_{md}[cbo]}{K_{md}+[cbo]}[cbo]\] \[V_{gh1-1}=\frac{V_{mg}[cbi]}{K_{mg}+[cbi]}[cbi]\] \[V_{glco}=\frac{V_{me}[glucose]}{K_{me}+[glucose]}[glucose]\]

Reactants, enzymes and parameters

Reactants	Meaning	Unit
cbo	Cellobiose extracellular	mM
cbi	Cellobiose intracellular	mM
glu	Glucose	mM
ethanol	Ethanol	mM

Table 2.1 Reactants in cellobiose pathway.

Enzymes	Meaning
transport	Transporter
gh1-1	β-glucosidase

Table 2.2 Enzymes of reactions in cellobiose pathway.

Parameters	Meaning	Value	Unit	Reference
\(C_{XS}\)	Bacteria carrying capacity	2.46	OD600	Estimated
\(V_{grwmax}\)	Maximum growth rate of yeast	0.1622	h^-1	Estimated
\(K_{mb}\)	Michaelis constant of transporter	3.5	μM	Lian, J., Li, Y., Hamedirad, M., & Zhao, H. (2014).
\(V_{mb}\)	Maximum transportation rate of transporter	0.08	μM/h	Lian, J., Li, Y., Hamedirad, M., & Zhao, H. (2014).
\(K_{mg}\)	Michaelis constant of gh1-1	880	μM	Chauve, M., Mathis, H., Huc, D., Casanave, D., & Ferreira, N. L. (2010).
\(V_{mg}\)	Maximum reaction rate of gh1-1	0.01	μM/h	Chauve, M., Mathis, H., Huc, D., Casanave, D., & Ferreira, N. L. (2010).
\(K_{me}\)	Michaelis constant of glycolysis	1.186*10⁶	mM	Rorke, D., & Kana, G. E. B. (2017).
\(V_{me}\)	Maximum reaction rate of glycolysis	0.1449	μM/h	Rorke, D., & Kana, G. E. B. (2017).

Table 2.3 Kinetic parameters of reactions in cellobiose pathway.

Simulation

We simulate cellobiose pathway in XX hours, using Simbiology Toolbox in MATLAB.

Figure 2.2 Simulation of cellobiose pathway.

However, we failed to fit the experimental data to simulation results for two reasons. First, some of kinetic parameters in model are not accurate. Though parameters are either obtained from literature or estimated by experiments, the standard error of some parameters are too big to be accurate. Second, we didn’t expect the secondary growth of S.cerevisiae, as well as the declining of ethanol, which is predicted to be reacted with acetic acid produced by glycolysis process.

Sensitivity Analysis

For cellobiose pathway, we did sensitivity analysis of the numerical integration of sensitivities, in order to investigate the rate-determined step of this pathway.

Figure 2.3 Numerical integration of sensitivities of kinetic parameters in 58 h.

According to the sensitivity analysis, the rate-determining steps (RDS) are the maximum reaction rate of the transportation property of transporter. The results are mirror to our wet lab for cellobiose pathway optimization.

Adhesion Platform

Overview

The aim to the adhesion platform is binding the E.coli to S.cerevisiaes to form a microbial collaboration platform utilizing streptavidin-biotin interaction. Given that we longed to get a better production rate of ethanol using this platform in our project, we made this model to :

1. simulate the kinetic process of adhesion platform, including the coculture-growth of E.coli and S.cerevisiae, the binding-dissociation process of these two organisms, using agent-based modeling (ABM);

2. prove that binding E.coli to S.cerevisiae can increase the production rate of ethanol using ABM;

3. define the Average Normalized Rate Constant (ANRC) to investigate the best experimental conditions for our lab.

Agent-based modeling

Two agent-based models (ABMs) were developed to investigate the adhesion platform, for ABM is a powerful type of model to simulate a complex system in stochastic way, as well as display the real-time animations of the simulated system.

In ABM, we can create thousands of agents and design them by several simple rules that we encoded in them, resemble to that we design the cell to form a device that could achieve some functions in synthetic biology, and set them in some conditions required. Then, we simulate them to see what would happen. This is an impressive method to explore how a complex group system evolves based on individual activity, or as called “emergence phenomenon“, so we use ABM as our major approach in this model.

We use NetLogo to build our ABMs. The source code could be found in our Supplementary and Coding.

Binding-Dissociation Model

This agent-based model (ABM) built in NetLogo shows us vivid results of how different types of cells behave in adhesion platform.

The diagram of simulation is as follows:

Figure 3.1 The diagram of binding-dissociation model.

Assumptions

To simplify the reality, several assumptions had been made as follows:
1. Agents are moving randomly obeying Brownian Motion in liquid environment.
2. Binding and dissociation process are considered independent events for SE_i(i≤5, SE_i is a S.cerevisiae cell attached with i E.coli cells.).
3. Adhesion platform does not affect the strength of streptavidin-biotin interaction.

Derivation of Assumption 2 and Calculations of each Probability

Derivation

Using parameters shown in Table 1, and simplifying the S.cerevisiae cell as a sphere, E.coli cell as a cylinder, we calculate the solid angle (Ω) of both of them.

\[\Omega=\iint_S{\sin\theta d\theta d\phi}\]

Therefore, the solid angle of a S.cerevisiae and an SE_i could be given.

\[\Omega_S=4\pi\] \[\Omega_{SE_i}\approx0.11\]

The proportion of occupied solid angle could be calculated.

\[\frac{\Omega_{SE_i}}{\Omega_S}\approx 0.96\%\lt1\%\]

Although the primary binding process may affect the secondary binding process, the proportion of occupied solid angle is less than 1%, which means assuming binding and dissociation process as independent events is reasonable. Independent events keep the same probability all the time, so that we could calculate the probabilities of both of them.

Binding Probability(Probb)

Given that binding only occurs when a SE_i meets with an E.coli cell, the sample space of the binding probability is considered as “A SE_i and an E.coli cell meeting each other”.

\[Probb=P("B"\mid"EM") P("EM")\]

where “B” equals “Binding” and “EM” equals “Effective meeting”. Effective Meeting, in other words, is the streptavidin and biotin meeting each other as two cells encounter. Therefore, alternative form if Probb is given.

\[Probb=\alpha \frac{N_{bio} A_{bio}}{A_{S.cere}}\cdot\frac{N_{strep} A_{strep}}{A_{E.co}}\]

By bringing parameters in Table 1, Probb is 54.503 % in our model.

Dissociation Probability(Probd)

Not like binding process, which may happen only when two cells encounter, the dissociation process could be taken place at any time, so the Probd is defined as the probability of dissociation in an unit time interval (Δt) following the exponential distribution.

\[Probd=1-{exp} (-k_d \Delta t)\]

By bringing parameters in Table 1, Probd is 0.339 % in our model.

*Supplementary: Derivation of Probd function
The dissociation process could be simplified as follows:

\[SE_i\to SE_{i-1}+E \]

And we could write down the ODE function of it.

\[\frac{-d[SE_i]}{dt}=k_d[SE_i]\]

By integrating it with \(t:0\to\Delta t\), \([SE_i]:[SE_i]_0\to[SE_i]_0+\Delta[SE_i]\), we have

\[\ln\left(\frac{[SE_i]_0-\Delta[SE_i]}{[SE_i]_0}\right)=-k_d\Delta t\]

Alternative form of this equation is as follows.

\[\frac{\Delta[SE_i]}{[SE_i]_0}=1-exp(-k_d\Delta t)\]

From the definition of Probd, we could find that

\[Probd=\frac{\Delta[SE_i]}{[SE_i]_0}\]

And we give the probability of dissociation.

\[Probd=1-exp(-k_d\Delta t)\]

Parameters and agents in ABM

Parameter name	Description	Value	Unit	Sources/Comments
S_E	Size of E.coli	2×0.5	μm	Kubischek HE(Jan 1990)
S_S	Size(radii) of S.cereivisae	3.75	μm	Walker K, Skelton H, Smith K(2002)
Ω_S	Solid angle of S.cereivisae	4π	-	Calculated in model
Ω_SE	Solid angle of SE_i	0.11	-	Calculated in model
Probb	Probability of binding process	54.503%	-	Calculated in model
Probd	Probability of biotin: streptavidin complex dissociation process	0.339%	-	Calculated in model
Probd_i	Probability of SE_i dissociation process	i=1 0.339% i=2 0.677% i=3 1.011% i=4 1.344% i=5 1.674%	-	Calculated in model
N_bio	Number of biotins displayed in S.cereivisae	16000	-	Parthasarathy, R., Bajaj, J., & Boder,E. T.(2005)
N_strep	Number of sreptavidin displayed in E.coli	160000	-	Park, M., Jose, J., Thömmes, S., Kim, J. I., Kang, M. J., & Pyun, J. C. (2011)
A_bio	Influential area per biotin	2.04×10^-3	μm²	PDB：4WF2
A_strep	Influential area per streptavidin	3.91×10^-5	μm²	Daniel, D. M., Drake, E. J., Hong, L. K., Gulick, A. M., & Sheldon, P. (2013)
A_S.cere	Surface area of a S.cerevisiae cell	176.72	μm²	Calculated in model
A_E.co	Surface area of a E.coli cell	6.29	μm²	Calculated in model
α	P("B"\|"EM")	1	-	*Assumed
Δt	Unit time interval in simulation	1	s	-
k_d	Dissociation rate constant	3.4×10^-3	s^-1	Wu, S. C., Ng, K. S., & Wong, S. L. (2009)
iniGal	Initial concentration of galactose	833	particles	1 particle=24mg/L galactose
iniS.cere	Initial concentration of S.cerevisiae in simulation		-	Variable:set to dfferent values in the model, 1 pixcor=1.2μm
iniE.coli	Initial concentration of E.coli in simulation		-	Variable: set to different values in the model
E_move	Average moving rate of E.coli cells		-	Variable: set to different values
S_move	Average moving rate of S.cerevisiae cells		-	Variable: set to different values
Y_E/g	E.coli-biomass yield	287.54	/particle	Estimated from experimental data
Y_S/g	S.cerevisiae-biomass yield	20.01	/particle	Estimated from experimental data
Size_world	The size of simulation world		world	1 world = 7.5×10^-3 μL

*Assumption is made due to the high affinity of streptavidin biotin interaction.

Table 3.1 Parameters used in ABM

Agent name	Description	Comments
E.coli	E.coli cell	-
SE_i	*A S.cerevisiae* cell attached with i E.coli cells**	i=(1,2,3,4,5)
galactose	Galactose particle	-

Table 3.2 Agents applied in ABM

Simulations

We focus mainly on two conditions to analysis our platform: the ratio of initial concentration of S.cerevisiae cells and E.coli cells (S:E), and initial concentrations of cells.

The ratio of initial concentration of S.cerevisiae cells and E.coli cells (S:E)

We change this ratio from 1:10 to 10:1 while keeping the initial total concentration of cells as a constant.

Here shows some of the simulation results:

Figure 3.2 Percentage of S.cerevisae attached with i E.coli cells (SE_i,i≤5) in total concentration of S.cerevisae in the simulation when initial ratio of S&E from 1:10 to 10:1 (line color from yellow to red).

Figure 3.3 Percentage of S.cerevisae attached with E.coli cells (the sum of SE_i,i≤5) in total concentration of S.cerevisae in the simulation when initial ratio of S&E from 1:10 to 10:1(line color from yellow to red)

The total concentration of cells

We change the initial total concentration of cells from 100 /world to 1000 /world while keeping the ratio of S.cerevisiae and E.coli to be 1:1. There are 10 different initial conditions and each condition was simulated 20 times.Note that the unit '/world' is the amount of cells per simulation world size, which is 7.5×10^-3μL.

Figure 3.4 Percentage of S.cerevisae attached with i E.coli cells (SE_i,i≤5) in total concentration of S.cerevisae in the simulation when total concentration of cell to be 500/world.

Figure 3.5 Percentage of S.cerevisae attached with E.coli cells (the sum of SE_i,i≤5) in total concentration of S.cerevisae in the simulation when initial concentration of all cells from 100 to 1000/world(line color from yellow to red).

Animations

As agent-based model could directly display the real-time behaviors of adhesion platform, animated simulations are given.

Note that the cycle-like agents with various colors are S.cerevisiae cells. The blue ones are SE₀, and those colors from pink to deep red are SE₁ to SE₅. The purple rod-like agents are E.coli cells, which are very tiny compared to S.cerevisiae cells. Galactose particles are not shown.

Animation 3.1 Animated simulation of S:E = 1 : 5 in 1000 min.

Animation 3.2 Animated simulation of S:E = 1 : 1 in 1000 min.

Animation 3.3 Animated simulation of S:E = 5 : 1 in 1000 min.

Production properties of SE_i

To theoretically explore the properties of ethanol-production rate of each SE_i (a single S.cerevisiae cell binding with i E.coli cells), we made some assumptions to simplify our problem.

Assumptions

1. The number of transporters of xylose is finite, stable.
2. The transporters of S.cerevisiae remain the their maximum transportation rate.
3. E.coli and S.cerevisiae are act as the xylose giver and xylose receiver without any other influence on each other.

Parameters and agents of the ABM

Parameter name	Description	Value	Unit	Sources/Comments
\(Y_{e/X}\)	Ethanol yield on xylose	48	mM/mM	Estimated from experimental data
\(\mu^{trsp}_{max}\)	Maximum uptake rate of transporter	0.16	mM/min	Estimated from experimental data
\(S_E\)	Size of E.coli	2×0.5	μm	Kubitschek HE (Jan 1990)
\(S_S\)	Size (radii) of S.cereivisae	3.75	μm	Walker K, Skelton H, Smith K (2002)
\(R_{diff}\)	Xylose and ethanol particles diffusion rate	-	pixcor/min	Set to different values in the model
\(R_{xp}\)	Xylose production rate	0.16	mM/min	*Assumed
\([xylose]_{bg}\)	Background concentration of xylose	81.72	mM	Estimated from experimental data

*An assumption made while lacking literature values and experimental values. We assumed this value to be consistent with \(\mu^{trsp}_{max}\) for simplification, and later found that this assumption will not influence the conclusions gained form the simulation results.

Agent Name	Description	Comments
E.coli	E.coli cell	-
SE_i	S.cerevisiae cell combined with i E.coli cells	i=(1,2,3,4,5)
Xylose	Xylose particle	1 particle=2E-3 mM
Ethanol	Ethanol particle	1 particle=2E-3 mM

Simulations

The simulations were set by these rules:
1. Xylose particles produced by E.coli cells and ethanol particles produced by S.cerevisiae cell are diffused by randomly moving —— Brownian Motion.
2. The simulation world was set closed in case that the agents would cycle through the world in NetLogo.
3. Each simulation was set to run 500 min for 20 times.
We set diffusion rate to be 0.1, 0.2, 0.3, 0.4 pixcor/s to simulate our model in separate conditions of SE₀, SE₁, SE₂, SE₃, SE₄, SE₅. Here shows some of the simulations. Note that 1 pixcor = 1.2μm

Figure 3.6 Ethanol producing simulations of SE_i (i=0, 1, 2, 3, 4, 5) in 500 min.

We normalized the ethanol production rate r_i as p_i, and p_i of SE_i were shown (i= 0, 1, 2, 3, 4, 5).Calculation method of r_i and p_i.

Figure 3.7 Normalized ethanol production rate (p_i) in diffusion rate of 0.1 pixcor/min.

Figure 3.8 Normalized ethanol production rate (p_i) in diffusion rate of 0.4 pixcor/min.

The results demonstrate that binding E.coli to S.cerevisiae could theoretically sharply increase the production rate for single cell, and we think there are two reasons for its effectiveness. The first reason is that the distance between the “giver” and “receiver” was sharply shorten so that the diffusion of xylose wouldn't be a limited factor anymore. The second is that, in micro-environment, as the binding number increases, the xylose produced by surrounding E.coli cells also increase, which offers S.cerevisiae cells the higher concentration of resource.

Animations

Two videos of simulations. One is from SE₀, and another is from SE₅:

Animation 3.4 Animated simulation of SE1 in 500 min.

Animation 3.5 Animated simulation of SE5 in 500 min.

Average Normalized Rate Constant (ANRC)

Introduction

The effectiveness of adhesion platform is not consistent with binding ratio of SE_n, which means a high binding ratio may not lead to high production rate of ethanol. For example, there are 1000 E.coli cells and 10 S.cerevisiae cells in adhesion platform, so the binding ratio may be very big, but the concentration of S.cerevisiae cells are too small to have a high production rate. So the effectiveness of adhesion platform should also consider the relative concentration of S.cerevisiae cells.

And this effectiveness is not just consistant with the production property of a single SE_i. It’s multiple interactions from a group of cells with different binding types, different binding ratios, different production properties. In other words, this is a complex system to be investigated its effectiveness.

So we define an equation named Average Normalized Rate Constant (ANRC) to analyze our simulation results, and give us a hint which experimental condition could lead to the best production.

\[ANRC=\frac{S}{S+E}\sum^n_{i=0}(p_i\times a_i)\]

S —— Concentration of S.cerevisiae cells.
E —— Concentration of E.coli cells.
p_i —— Normalized production rate constant of each SE_i.
α_i —— Proportion of SE_i. \(\alpha_i=\frac{SE_i}{S}\times 100\%\)

And the difference between adhesion platform and non-adhesion platform could be defined as their difference of each ANRC (Δ), which would tells us whether our system is better than non-adhesion platform.

\[\Delta=ANRC_{adhe}-ANRC_{non-adhe}\]

ANRC and Δ contained two components. One is the results gained from “Binding-Dissociation Model”, which are S, E, α_i, another is the results gotten from “Production Properties of SE_i”, which are values of p_i. So this function is the combination analysis of both binding-dissociation and ethanol production.

Supplementary: Derivation of ANRC and Δ

Derivation of ANRC

By adopting the Monod function and stoichiometry rules in our adhesion platform, the production could be simplified as:

The production rate of ethanol is given.

\[V_{eth}=\frac{d[ethanol]}{dt}=\sum^n_{i=0}SE_i\left(\frac{k_i[xylose]}{K_i+[xylose]}\right)\]

The alternative form of this function is:

\[V_{eth}=S\times\sum^n_{i=0}\alpha_i\left(\frac{k_i[xylose]}{K_i+[xylose]}\right)\]

If we consider \(v_{eth}\) as the addition of i \(v_{ethi}\), we would get

\[V_{eth_i}=S\alpha_i\times\left(\frac{k_i[xylose]}{K_i+[xylose]}\right)\]

We could find that the production rate constant of \(v_{ethi}\) is the latter item in function, which we defined as \(r_i\).

\[r_i=\left(\frac{k_i[xylose]}{K_i+[xylose]}\right)\]

This means the factors determining the rate constant of each type of SE_i are (1) \(k_i\),\(K_i\) the characteristics of SE_i which are the same for all type of SE_i, because their ethanol production unit are S.cerevisiae cell. (2) [xylose] the partial concentration of the xylose surrounding each cell, and this is the difference between different types of SE_i, because the partial concentrations of SE_i is higher than that of SE_i-1.

The partial concentration of xylose for SE_i could be divided into two parts —— a part produced from E.coli cells binding it ([xylose]_self), and a part from other E.coli cells as background ([xylose]_bg).

\[[xylose]=[xylose]_{self}+[xyloes]_{bg}\]

[xylose]_bg is the stable concentration of xylose in this system, and is determined by its production rate and consumption rate. Easy to notice that if [xylose]_bg is high enough, the concentration of xylose ([xylose]) would be nothing different between adhesion platform and non-adhesion platform, which means the advantage of adhesion platform would not be so significant. Therefore, to overcome it, one possible solution is to accelerate the process of consuming xylose by increasing the transportation rate of the transporters on S.cerevisiae cells to decrease the [xylose]_bg. It seems that the transporter is the key factor of making adhesion platform significant.

[xylose]_self is the characteristic property of each type of SE_i that could not be affected by environments. Easy to find that simulations in “Production Properties of SE_i” are to get r_i values by setting the concentration of S.cerevisiae cells to be 1/world.

By normalizing the r_i values to the maximum of them, we have the normalized production rate constant p_i.

\[p_i=\frac{r_i}{max(r_i)}\]

So the normalized ethanol production rate is as follows:

\[[Normalized_{V_{eth}}]=S\times\sum^n_{i=0}(\alpha_i\times p_i)\]

To eliminate the distinctions resulting from different concentration of cells in different systems, we take the average of the [Normalized v_eth] to obtain the mean level of ethanol production rate per cell, and defined it as Average Normalized Rate Constant (ANRC).

And the equation of ANRC is also given:

\[ANRC=\frac{S}{S+E}\sum^n_{i=0}(p_i\times \alpha_i)\]

Derivation of Δ

Wondering whether our adhesion platform would have a better performance than non-adhesion platform, we defined Δ.

\[\Delta=ANRC_{adhe}-ANRC_{non-adhe}\]

From the definition of ANRC, easy to get \(ANRC_{non-adhe}\) .

\[ANRC_{non-adhe}=p_0\times\frac{S}{S+E}\]

Then the function of Δ is as follows:

\[\Delta=\frac{S}{S+E}\sum^n_{i=0}[(p_i-p_0)\times \alpha_i]\]

Analysis of adhesion platform

By changing the conditions in “Binding-Dissociation Model”, we calculated ANRC and Δ of each simulations. Results are shown as follows.

Note that the blue curve is the mean level of ANRC and Δ values.

Figure 3.9 ANRC values of different initial ratios of S.cerevisiae and E.coli.

Figure 3.10 Δ values of different initial ratios of S.cerevisiae and E.coli.

Figure 3.11 ANRC values of different initial concentrations of cells.

Figure 3.12 Δ values of different initial concentrations of cells.

Given that Δ values are bigger than zero in our simulations, we now can conclude that our adhesion platform would get a better performance in ethanol production than non-adhesion platform.

By analysis above, we can conclude that the best initial ratio of S.cerevisiae and E.coli is 1 : 2. This conclusion is mirrored to the wet lab, and is meaningful for optimizing our project.

Reference

Xylose Pathway
[1] Eliasson, A., Hofmeyr, J. H. S., Pedler, S., & Hahnhägerdal, B. (2001). The xylose reductase/xylitol dehydrogenase/xylulokinase ratio affects product formation in recombinant, xylose-utilising saccharomyces cerevisiae. Enzyme & Microbial Technology, 29(5), 288-297.
[2] Richard, P., Toivari, M.H., Penttila, M., 2000. The role of xylulokinase in Saccharomyces cerevisiae xylulose catabolism. FEMS Microbiol Lett, 190, 39-43.
[3] Rizzi, M., Erlemann, P., Buithanh, N.A., Dellweg, H., 1988. Xylose Fermentation by Yeasts .4. Purification and Kinetic-Studies of Xylose Reductase from Pichia-Stipitis. Applied Microbiology and Biotechnology, 29, 148-154.
Cellobiose pathway
[1] Lian, J., Li, Y., Hamedirad, M., & Zhao, H. (2014). Directed evolution of a cellodextrin transporter for improved biofuel production under anaerobic conditions in saccharomyces cerevisiae. Biotechnology & Bioengineering, 111(8), 1521-31.
[2] Chauve, M., Mathis, H., Huc, D., Casanave, D., Monot, F., & Ferreira, N. L. (2010). Comparative kinetic analysis of two fungal β-glucosidases. Biotechnology for Biofuels,3,1(2010-02-11), 3(1), 3.
[3] Rorke, D., & Kana, G. E. B. (2017). Kinetics of bioethanol production from waste sorghum leaves using saccharomyces cerevisiae by4743. Fermentation, 3(2), 19.
Adhesion platform model
[1] Kubitschek, H. E. (1990). Cell volume increase in escherichia coli after shifts to richer media. Journal of Bacteriology, 172(1), 94.
[2] Walker, K., Skelton, H., & Smith, K. (2002). Cutaneous lesions showing giant yeast forms of blastomyces dermatitidis. Journal of Cutaneous Pathology, 29(10), 616.
[3] Parthasarathy, R., Bajaj, J., & Boder, E. T. (2005). An immobilized biotin ligase: surface display of escherichia coli bira on saccharomyces cerevisiae. Biotechnology Progress, 21(6), 1627–1631.
[4] Park, M., Jose, J., Thömmes, S., Kim, J. I., Kang, M. J., & Pyun, J. C. (2011). Autodisplay of streptavidin. Enzyme & Microbial Technology,48(4–5), 307-311.
[5] Demonte, D., Drake, E. J., Lim, K. H., Gulick, A. M., & Park, S. (2013). Structure‐based engineering of streptavidin monomer with a reduced biotin dissociation rate. Proteins-structure Function & Bioinformatics,81(9), 1621.
[6] Wu, S. C., Ng, K. K., & Wong, S. L. (2009). Engineering monomeric streptavidin and its ligands with infinite affinity in binding but reversibility in interaction. Proteins-structure Function & Bioinformatics, 77(2), 404-412.

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+       </button>
+		<a href="https://2017.igem.org/Team:OUC-China"><img src="https://static.igem.org/mediawiki/2017/8/89/T--OUC-China--logo.jpg" height="50px"/></a></div>
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 		    	<div><a href="https://2017.igem.org/Team:OUC-China/Description" style="line-height: 40px; " class="ouc-navdown">Description</a></div>
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-<div class="ouc-right">
+                <div class="ouc-right">
-		    	    <a style="line-height: 40px; font-size: 15px;" class="ouc-navdown">Proof of concept</a>
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 		    	    	<a href="https://2017.igem.org/Team:OUC-China/proof1" style="display: block">Basic fermentation</a>
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-		    	    	<a href="https://2017.igem.org/Team:OUC-China/proof3" style="display: block"> MINI-GRE </a>
+		    	    	<a href="https://2017.igem.org/Team:OUC-China/proof3" style="display: block">MINI-GRE</a>
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      </p>
      <p style="font-size: 20px">
-     	Firstly, we built pathway models (xylose pathway and cellobiose pathway) in MATLAB using SimBiology Toolbox by a set of enzymatic reaction kinetic equations in order to confirm our pathway design of producing ethanol. And we did sensitivity analyses using local methods to explore the rate-determined step in our pathways, which would further help us optimize the pathways for higher production.
+     	Firstly, we built pathway models (xylose pathway and cellobiose pathway) in MATLAB using Simbiology Toolbox by a set of enzymatic reaction kinetic equations in order to confirm our pathway design of producing ethanol. And we did sensitivity analyses using local methods to explore the rate-determined step in our pathways, which would further help us optimize the pathways for higher production.
      </p>
-     <p style="font-size: 20px">
+     <p style="font-size: 20px" id="sup">
      	Secondly, we built agent-based models (ABM, a wonderful type of models to investigate complex dynamic systems) using Netlogo. We utilized these models to run real-time simulations of adhesion platform we built, as well as explore the properties of the system. This allowed to show us vividly how would the system behave (e.g. coculture growth, binding ratios over time), and to prove our design.
      </p>
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      	Our models are modefied by our wet lab date and gave a wonderful guidance to our experiment in return.
      </p>
+<p style="font-size:20px" ><a href="https://static.igem.org/mediawiki/2017/1/11/T--OUC-China--sup.zip"><i>Supplementary and Coding could be download here</i></a></p>
      <br/><br/><br/><br/>
      <!--======================================================XYLOSE=============================================================-->
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                          <h3 class="ouc-heading" style="color: #66BCC7"><strong>Simulation</strong></h3>
                          <P style="font-size: 20px">
-                         	We simulated xylose pathway in 90 hours, using Simbiology Toolbox in MATLAB. The simulation file can be download in Github.
+                         	We simulated xylose pathway in 90 hours, using Simbiology Toolbox in MATLAB. The simulation file can be download in <a href="#sup">Supplementary and Coding</a>.
                          </P>
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              </p>
              <p style="font-size: 20px">
-             	We use NetLogo to build our ABMs. The source code could be found in our Github.
+             	We use NetLogo to build our ABMs. The source code could be found in our <a href="#sup">Supplementary and Coding</a>.
              </p>
          <h3 class="ouc-heading" style="color: #66BCC7"><strong>Binding-Dissociation Model</strong></h3>
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-                     			<td>Size<sub>world</sub></td> <tdThe size of simulation world</td> <td></td> <td>world</td> <td>1 world = 7.5×10<sup>-3</sup> μL</td>
+                     			<td>Size<sub>world</sub></td> <td>The size of simulation world</td> <td></td> <td>world</td> <td>1 world = 7.5×10<sup>-3</sup> μL</td>
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      	                    	<img src="https://static.igem.org/mediawiki/2017/a/aa/T-OUC-China--model3-2.png" height="300px"/><br/>
-				                    Figure 3.2 Percentage of <i>S.cerevisae</i> attached with i <i>E.coli</i> cells (SE<sub>i</sub>,i≤5) in total concentration of <i>S.cerevisae<i> in the simulation when initial ratio of S&E to be 1:1.
+				                    Figure 3.2 Percentage of <i>S.cerevisae</i> attached with i <i>E.coli</i> cells (SE<sub>i</sub>,i≤5) in total concentration of <i>S.cerevisae</i> in the simulation when initial ratio of S&E from 1:10 to 10:1 (line color from yellow to red).
                              </p>
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      	                    	<img src="https://static.igem.org/mediawiki/2017/0/0f/T--OUC-China--model3-3.jpg" height="300px"/><br/>
-                                     Figure 3.3 Percentage of <i>S.cerevisae</i> attached with i <i>E.coli</i> cells (SE<sub>i</sub>,i≤5) in total concentration of <i>S.cerevisae<i> in the simulation when initial ratio of S&E from 1:10 to 10:1(line color from yellow to red)                             </p>
+                                     Figure 3.3 Percentage of <i>S.cerevisae</i> attached with  <i>E.coli</i> cells (the sum of SE<sub>i</sub>,i≤5) in total concentration of <i>S.cerevisae</i> in the simulation when initial ratio of S&E from 1:10 to 10:1(line color from yellow to red)                             </p>
                              </div>
                          </div>
                          <h4><strong>The total concentration of cells</strong></h4>
 						<p style="font-size: 20px">
-							We change the initial total concentration of cells from 100 /world to 1000 /world while keeping the ratio of <i>S.cerevisiae</i> and <i>E.coli</i> to be 1:1.  There are 10 different initial conditions and each condition was simulated 20 times.<strong>Note that the unit '/world' is the amount of cells per simulation world size, which is 7.5×10<sup>-3</sup>μL。</strong>
+							We change the initial total concentration of cells from 100 /world to 1000 /world while keeping the ratio of <i>S.cerevisiae</i> and <i>E.coli</i> to be 1:1.  There are 10 different initial conditions and each condition was simulated 20 times.<strong>Note that the unit '/world' is the amount of cells per simulation world size, which is 7.5×10<sup>-3</sup>μL.</strong>
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      	                    	<img src="https://static.igem.org/mediawiki/2017/c/cf/T--OUC-China--mo3-4.png" height="300px"/><br/>
-				                    Figure 3.4 Percentage of <i>S.cerevisae</i> attached with i <i>E.coli</i> cells (SE<sub>i</sub>,i≤5) in total concentration of <i>S.cerevisae<i> in the simulation when total concentration of cell to be 500/world.
+				                    Figure 3.4 Percentage of <i>S.cerevisae</i> attached with i <i>E.coli</i> cells (SE<sub>i</sub>,i≤5) in total concentration of <i>S.cerevisae</i> in the simulation when total concentration of cell to be 500/world.
                              </p>
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                              <p style="color: gray;text-align: center;">
      	                    	<img src="https://static.igem.org/mediawiki/2017/c/c4/T--OUC-China--mo3-5.png" height="300px"/><br/>
-                                     Figure 3.5 Percentage of <i>S.cerevisae</i> attached with i <i>E.coli</i> cells (SE<sub>i</sub>,i≤5) in total concentration of <i>S.cerevisae<i> in the simulation when initial concentration of all cells from 100 to 1000/world(line color from yellow to red).                               </p>
+                                     Figure 3.5 Percentage of <i>S.cerevisae</i> attached with  <i>E.coli</i> cells (the sum of SE<sub>i</sub>,i≤5) in total concentration of <i>S.cerevisae</i> in the simulation when initial concentration of all cells from 100 to 1000/world(line color from yellow to red).                               </p>
                              </div>
                          </div>

Difference between revisions of "Team:OUC-China/Model"

Latest revision as of 14:00, 18 November 2017

Model

Overview

Xylose Pathway

Overview

Assumptions

Reactions

ODEs

Reactans, enzymes and paramenters

Simulation

Sensitivity Analysis

Cellobiose pathway

Overview

Assumptions

Reactions

ODEs

Reactants, enzymes and parameters

Simulation

Sensitivity Analysis

Adhesion Platform

Overview

Agent-based modeling

Binding-Dissociation Model

Assumptions

Derivation of Assumption 2 and Calculations of each Probability

Derivation

Binding Probability(Probb)

Dissociation Probability(Probd)

Parameters and agents in ABM

Simulations

The ratio of initial concentration of S.cerevisiae cells and E.coli cells (S:E)

The total concentration of cells

Animations

Production properties of SEi

Assumptions

Parameters and agents of the ABM

Simulations

Animations

Average Normalized Rate Constant (ANRC)

Introduction

Supplementary: Derivation of ANRC and Δ

Derivation of ANRC

Derivation of Δ

Analysis of adhesion platform

Reference

Production properties of SE_i