Modeling

Abstract

Our project mainly focuses on designing gene circuits to gather copper ions,degrade cyanide,
detoxify fluoride and suggest whether the water is safe for further use. With these giant goals, the
first thing we needed to do is using computational method simulate the biological process and figure
out whether our design is feasible. We established DDEs (delay differential equations) to see whether our
instructors are trustable and give some further information for the detective part of our toolkit. Then we tested the robustness and sensitivity to get a broader insight of biological system both in single cell level and multicellular level. By doing this, we can get their properties for better application.

Single Cell Level

DDEs Simulation

There are two kinds of E. Coli in the project— E. worker and E. instructor. The former ones produce some proteins binding with copper ions in the polluted water and the latter ones tell us whether the water is safe enough for further use. Since the thing we care about most is the safety of the water and the workers will be dedicated to remove the ions in the water before we decided to let them flow to the following pool, we established some equations to simulate the biological process of instructors. Considering about it will take some time for the transcription and translation process before a protein can bind with some certain promoters, we use DDEs instead of ODEs to make our simulation closer to the reality. And here are the equations:

Table 1: Descriptions and Values of the Parameters.

parameter	description	value	reference
copynum	copy number of pACYDuet-1 plasmid	18~22	[1]
trc1	transcription rate of mCII	w/o inducing: 0 inducing: 1	[2]
deg1	transcription rate of mCII	0.12	[3]
deg2	degradation rate of CII	0.1	[4]
deg3	degradation rate of mCI	0.12	[3]
deg4	degradation rate of CI	0.042	[4]
deg5	degradation rate of mGFP	0.13	[5]
deg6	degradation rate of GFP	0.017	[6]
deg7	degradation rate of mRFP	0.13	[5]
deg8	degradation rate of RFP	0.017	[6]
trl1	translation rate of CII	0.12	[4]
trl2	translation rate of CI	0.09	[3]
trl3	translation rate of GFP	5.4	[7]
trl4	translation rate of RFP	5.4	[7]
Vmax1	maximum transcription rate when induced by CII protein	0.9	[4]
Vmax2	maximum transcription rate when induced by CI2 protein	0.66	[4]
τ1	time for CII transcription, translation and folding	0.24min	estimated the same as CI2
τ2	time for CI2 transcription, translation and folding	0.24min	[3]
km1	apparent association constant for CII binding with pRE promoter	0.398	[8]
km2	apparent association constant for CI2 binding with pR promoter	1.58*10-3	[3]
k1	reaction constant for CI forming CI2	3	[4]
k2	reaction constant for CI2 disassociating to CI	30	[4]

The results of simulation are shown in the graphs below:

Fig.1,2: Simulation Results.

As you can notice in the picture, the expression level of fluorescent protein changes a lot from polluted and non-polluted water. Thus, by detecting the fluorescence intensity of each protein, we can gain the information about whether the water is safe for further use. Considering about the severe consequences of taking in excessive amount of copper ions, we should make sure that our data is credible and the information we get is accurate.

Fig.3: The Dealing Process.

We simulated the whole process of the water-dealing procedure. In the view of that the transcription rate of the copper sensitive promoter is related to the concentration of copper in the water, we divided the treating process into several parts with different transcriptional rate, and combined all the data eventually to make our simulation closer to the reality. The result showed below indicates that only detecting one of the fluorescent intensity only is enough to get the information we want. But we should always detect the other fluorescent intensity redundantly to make the conclusion more trustable.

Fig.4: The Overview Graph of Continuous Dealing Process.

Robustness and Sensitivity Analysis

Considering there are so many parameters and variables in the equations, we chose to use numerical solutions to analyze the robustness of the equations. Since trc₁, τ₁ and τ₂ are three changeable parameters that may contribute most to the output, we decided to put our focus on these three parameters in this part. Here are some graphs representing the expression states under different trc₁ values.

Fig.5: The Expression States with Different trc₁ Values.

From left to right, up to down, the values of trc₁ are: 0, 0.002, 0.005 and 1. These four graphs represent four different stable states with different trc₁ values. Although the final expression levels are different, they all achieve a stable state. In other words, we can judge whether the water is safe for further use just by detecting limited numbers of data.

Then we shifted our focus onto the specific parameters. The first thing we did is to analyze how the expression levels of GFP and RFP are sensitive to the value of trc₁. We changed the value of trc₁ from 0 to 0.1 at the step length of 0.001. The picture shows below is the simulating result.

Fig.6: Parameter Sweep.

As you can see in the picture, the green curve represents the GFP expression condition, and the red one represents that of RFP. The fewer and more scattered the curves are, the faster the final output changes. Based on the picture showed above, we can conclude that the expression of GFP and RFP is not so sensitive to trc₁ with increasing value of it from 0 to 1. When trc₁ approaches to ∞, the expression level of GFP is close to the black curve while that of RFP is close to the blue one.

To test the effects of the expression condition on GFP and RFP caused by trc₁ a step further, we pictured the fluorescent intensity of GFP and RFP at t equaling 200min (an estimated stable state) under different trc₁ values.

Fig.7,8: Robustness and Sensitivity Analysis.

Table 2: The Digital Number Changes of GFP and RFP When trc₁ Changes At the Step Length of 0.01.

trc1*10-2	[0 1]	[1 2]	[2 3]	[3 4]	[4 5]	[5 6]	[6 7]	[7 8]	[8 9]	[9 10]
GFP	2332.70	11.86	3.33	1.71	1.08	0.76	0.57	0.45	0.36	0.30
RFP	423.87	282.56	201.85	151.43	117.81	94.28	77.16	64.32	54.44	46.68

The left picture above shows the fluorescent intensity of GFP and RFP on stable states under different trc₁ values, the right one shows the rate of change. The chart above shows the digital number changes of GFP and RFP when trc₁ changes at the step length of 0.01.

According to the data and graphs above, we will find it’s not hard to make the conclusions below:

1. The expression level of GFP and RFP is rather sensitive to trc₁, when the value of trc₁₁ becomes larger, this kind of influence reduces and will finally have no effect on the expression level of GFP and RFP. (In the extreme state)

2. The influence of trc₁ to the expression condition of GFP is short but obvious. When the value of trc₁ changes between 0 and 0.01, the change of relative fluorescent intensity of GFP is about 2322. While that of trc₁ changes between 0.09 and 0.1, the change of relative fluorescent intensity of GFP is only about 0.3, which is close to 0. In addition, the graph on the right shows that the rate of change of the GFP fluorescent intensity reaches the maximum to at site A and reduces tremendously to nearly 0.

3. The influence of trc₁ on the expression of RFP is mild but endless. When the value of trc₁ changes between 0 and 0.01, the change of relative fluorescent intensity of GFP is about 424. When that of trc₁ changes between 0.09 and 0.1, the change of relative fluorescent intensity of RFP is about 47. The fluorescent intensity of RFP is still increasing when the value of trc₁ equals 0.1. Actually, this kind of increase will keep on even when the value of trc₁ is over 1.

4.At site 1 on the left graph, the value of trc₁ is 0.004, where the fluorescent intensity of GFP and RFP is exactly the same. And at site B on the right graph, the value of trc₁ is about 0.0099, where the rate of change of the fluorescent intensity of GFP and RFP is the same.

Considering that the treating process is a long-lasting period, during which the concentration of copper ions changes a lot, the transcription rate varies from time to time. We tested the time that would consume to reach a new stable state with the change of trc₁.

Fig.9: Time Needed to Reach A New Stable State with Different trc₁.

This picture represents the value of trc₁ changing from 0.005 to 0.002. We defined that when the related error is equal to or less than 5%, it reaches a stable state. So, as you can see in the picture above, when t equals 373 (site E), the fluorescent intensity of GFP and RFP are identical. When t equals 554 (site F), the expression level of RFP reaches the stable state. And when t equals 646 (site G), the expression level of GFP reaches the stable state.

The above is what we did to analyze the output sensitivity to trc₁. Then we tested how the parameter τ (time delay) affects the output result. We changed the value of τ from 0.12 to 0.36 at the step length of 0.01 to see the result.

Fig.10,11: The Effection of τ to the Output.

As the picture shown above, the fluorescent intensity of GFP and RFP is extremely insensitive to the change of τ (time delay). When the parameter τ (time delay) changes from 0.12 to 0.36, the fluorescent intensity of GFP in the stable state only changes at the value of 5.28, while that of RFP only changes at the value of 2.23.

Circuit Improvement

As mentioned above, that our system is rather sensitive to the concentration of copper ions. However, the detected results may fail to tell what the concentration of cooper ions is exactly. If the basic transcriptional level of the promoter is higher than 0.005, the expression level of RFP is always comparatively much higher than that of GFP. Unfortunately, according to the results from our wet lab, the promoter we chose at the first time has a quit severe leakage. Thus we will have no idea about whether the water is safe enough for further use by detecting the fluorescent intensity. Since it’s the expression level of RFP that is higher than our expectations, we tried to reduce it by improving our gene circuits. According to our designing, it’s CII protein that induces the expression of RFP. So we believe that adding a degradation tag may be a good solution. Then we tested how many times of the original degradation rate is needed. And here is the result of the transcription rate where the water is safe according to the national standard. Considering about that there are always some oscillations of parameters in the biological system, we also did a gradient analysis to see whether these oscillations may affect the result and whether there are any superior choices.

Fig.12,13: Circuit Improvement.

As you can see in the picture above, a specific times of the degradation rate is required. However, there is no much room for parameter oscillations. Since the degradation rate cannot be predicted accurately in vivo, especially when some tags are added (the accelerated degradation rate largely depends on the proteins inside cells), we tried to find some other promoters that are more suitable for our project.

Luckily, we found the PpcoA promoter. As you can see in the graph below, there is no significant difference in the relative fluorescent intensity between the pET28a plasmids containing and not containing the PpcoA promoter. So it can be predicted that the transcription rate of this promoter can meet our need.

Fig.14: Promoter Test.

Multicelluar Level

Environmental factors

Since the wastewater we tried to deal with come from the process of industrial producing, we must consider some environmental factors that can affect the treatment procedure. There are also some other ions in the water that can form some chelate compounds combining with copper ions. And these ions have a competitive relationship with copper-binding proteins. By analyzing the whole environmental surroundings, we can get the information about how many copper-binding proteins we need, thus to calculate how many bacteria we should paint on the surface of the RBC (Rotating Biological Contactor). According to previous literature review, we found the major existing forms of copper ions in the wastewater from industries are [Cu(CN)3]^2- and [Cu(NH3)4]²⁺. There are 4 major reactions occur in the water, and here are the equations.

After looking up some papers, we found the concentration of [Cu(CN)₃]^2- is estimated to be 50mg/L^[9] and that of [Cu(NH3)4]²⁺/ is about 20.55mM^[10] in industrial wastewater. Since our E. Kungfu can also oxidize cyanide, CN^- can be removed from the water, moving the balance of the process towards the direction in favor of degrading [Cu(CN)₃]^2-. Thus, most of [Cu(CN)₃]^2- will be transformed into Cu⁺. Then disproportionation reaction occurs, all Cu⁺ will be transformed into either Cu or Cu²⁺. As for [Cu(NH₃)4]²⁺, since one of the degradation products is NH₃, which is a kind of gas that will be released from water, the balance of the process will also move towards the direction in favor of degradation. So the number of CBP we need is just the exact amount of copper ions that we try to remove from the wastewater. (The apparent dissociation constants for Cu(I)-binding proteins and ligands of low-mass are all about 10¹⁵ and we will constantly remove bacteria containing CBPs from the wastewater. So, we assumed that all the copper ions can be adsorbed by these copper binding proteins.) According to the data mentioned above, the number of CBPs we need to make wastewater meet the national standard is about 3.09*10²¹. By estimating how many copper binding proteins an E. Coli can bear, we can gain the data of the number of E. Kungfu we need to plant onto the rotating disks. And by combining with the diameter of the rotating disk, we can know how thick of the cultural medium should be needed to fixate enough E. worker.

RBC Toolkit

We also planned to optimize some parameters of the rotating biological contactors (RBC) to achieve the best function. Although some articles claimed that increase the speed of the rotating disks can make the whole dealing system more efficient. We must consider that with the increase of the speed, the enormous energy consumption and the burden of spindle will be a big problem. This is why we should choose a suitable speed before we put our idea into application. The equations below describe the relationship between some critical parameters, according to which we can adjust the rotating speed.

Table 3: Parameter Descriptions

parameter	description	unit
A	the overall area of rotating disks	m2
Q	the volume of the sewage needed to be treated	m3/d
S0	the BOD5 value in the sewage	mg/L
LA	the BOD5 consuming by bacteria per square meter, per day	g/(m2d)
m	the pieces of the rotating disks	piece
D	the diameter of the rotating disks	m
L	the effective length of the tank	m
a	the net space between contiguous disks	m
b	the thickness of the rotating disks	m
K	coefficient	/
V1	net effective volume	m3
δ	coefficient	m
n0	rotating speed	r/min
Q1	the setting volume of the tank	m3/d

Since we cannot get most values of the parameters in real conditions, we failed to specify these equations in the form of some graphs. In the future, we may try to find a balance.

References

[1] Citation needed
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[3] Kinetics of lambda phage manipulate genes, regulatory networks and protein interaction lysogenic state / cleavage transition characteristics. Hui Ding, et al. Journ al of Inn er Mongolia University Sep. 2007 Vol. 38 No. 5
[4] Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage l-Infected Escherichiacoli Cells Adam Arkin ,et al. Genetics 149: 1633–1648(August 1998)
[5] Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays, Jonathan A. Bernstein, et al. PNAS July23, 2002, vol.99 no.15, 9697–9702
[6] http://bionumbers.hms.harvard.edu/
[7] https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters
[8] Kinetic analysis of mutations affecting the cII activation site at the PRE promoter of bacteriophage λ, MING-CHE SHIH, et al. Proc. Natl. Acad. Sipi. USA, Vol. 81, pp. 6432-6436, October 1984, Genetics
[9] The research of treating sodium hypochlorite containing [Cu(CN)₃]^2- complex ions in wastewater. Shiqian Wei Vol 26 No. 5 Xuchang University Vol. l26. No. 5, sept. 2007
[10] The research of treating copper ammonia complex in wastewater use TMT. Dongmei Liao, Chinese Water Supply and Drainage Vol. 22, Sept. 2006