Modelling

Intro

In an effort to understand our system and understand the dynamics behind our construct and its impact on the bacteria we decided to make a computer simulation. With this simulation we hope to be able to predict the amount of inulin produced by our bacteria and maybe get a measurement of how much sucrose is needed to meet the inulin requirements for health benifits.

Disclaimer

Due to the limited characterization we’ve had with our enzymes it hasn’t been possible to make any exact modelling but we hope to display some of the basic functionality of our bacterial system. With this model we hope to show the dynamics behind the process of inulin metabolization but due to the large number of variables we have had to assume a lot of parameters.

For the modelling purpose we’ve used the program Berkeley Madonna which is a simple yet effective program to model biological systems. (http://www.berkeleymadonna.com/)

Theory

SST: is able to cleave and combines a sucrose with a fructose moiety of a cleaved sucrose molecule to make the trisaccharide 1-kestose through a beta2->1 linkage.

FFT: uses 1-kestose as a substrate and elongates this by the iterative addition of n amount of fructose moiety again through beta2->1 linkages. This enzyme also has the ability to cleave sucrose into its two constituents.

Figur 1 – In the image we can see the basic structure of inulin, with the bottom part being the first sucrose and the “arm” extending upward from the green fructose are fructose units added by FFT. (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC196615/)

When making a model it is often helpful to make a simple sketch of your system to visualize it and then figuring out what goes into your system where, and what comes out. It is also essential to find bottlenecks i.e. where there might be delays in the system due to enzyme inefficiencies or transport of precursors across membranes etc. From this sketch we constructed a causality diagram which is shown below.

In this diagram we postulate that SST and FFT work in cooperation, that is, they are not independent of each other. In the diagram sucrose enters the system from an infinite pool visualized as a cloud. The reason for this is that we assume that for each individual bacteria the amount of sucrose available seems infinite, but in reality this would not be the case.

Sucrose is then cleaved into its two constituent parts; sucrose and glucose which both SST and FFT are capable of. In our model system this is represented by XXT which is the combined catalytic cleavage of sucrose by both enzymes. The reason for this approach is because the amount of fructose inside the cell is the major rate limiting factor for inulin production so we chose this way of modelling it due to it being easier to work with even though this is not what happens in real life. Because the inulin is primarily made up of fructose this creates a large excess of glucose. The way we handled this is to assume the bacteria is able to use the glucose in its own metabolism, which in the model is called Bak.

SST catalyzes the formation of the trisaccahride 1-kestose from a sucrose and a fructose unit. FFT catalyses the addition of several more fructose units in an iterative loop that terminates when the chain has reached an adequate length. In our model we chose a chain length of 5 as the end product, this is modelled as a loop function in our model.

Promoters

For our desired construct we wanted to be able to control transcription of our SST and FFT genes by both having a sucrose sensitive promoter and a heat sensitive promoter. For the model we have made three different types of promoters to simulate our desired final construct. In our model the promoters activate the SST and FFT activities.

The first promoter is a simple constitutive promoter that is always active and thereby always activates transcription. The reason for this promoter was to simply show that the system could work.

The second promoter is a heat shock promoter that only activates transcription when the heat of the system is sufficiently high. The thought behind this approach is if we wanted to have our bacteria in a yogurt that is to be stored in a refrigerator and then when it enters the stomach of the host it heats up and activates transcription of our genes. In the simulation we have simply modelled the heat shock promoter after a delay function, so that after five seconds we go from cold to hot and transcription proceeds.

The last promoter is a sucrose sensitive promoter that is activated by high sucrose concentrations. The way we have modelled it, is a basic Michaelis-Menten model that is dependent on the sucrose concentration.

Result from the Model

Plot of the sucrose sensitive promoter activity (black) and sucrose concetration (red)

As the promoter activates transcription of the SST and FFT genes we see the Fructose (blue) and Glucose (green) concentrations rise

The fructose and glucose is converted to trisaccharide (orange) in a 2 to 1 ratio by SST

The trisaccharide is converted into inulin1 (pink) by the addition of a fructose moiety.Through the iterative additon of more and more fructose units to the inulin chain we observe the elongation seen first as inulin2(blue) then inulin3(black) and inulin 4(red). Please note the time delay between the tops of the trisaccharide and inu1-4 graphs, this indicates that they are causally linked.

Finally we see the end product Inulin5 (yellow line) that accumulates as the system eats up the continuously added sucrose. What is noteworthy is that even though there is an initial spike in the concentrations of sucrose, fructose, glucose and all the inulins they all level out eventually (except for inulin5) this inducates that the system has found an equlibrium and now functions as an inulin generator.

Discussion

Is our model sufficient to predict the behaviour of our construct? There are too many variables in a biological system to make any real simulation but we can make some predictions on the behaviour. Our model contains approximately 27 different parameters that are adjustable. With this being a model we simply started out by having all the constants set to either 1 or 0 but this is way too simplified and resulted in some strange behaviour.

By fine tuning certain parameters such as the enzyme decay rate or how hungry the bacteria are for glucose we can adjust the plots to fit certain behaviour, but seeing as we haven’t gotten much data from our practical characterization experiments it is nearly impossible to find a perfect fit.

For example we could argue that the excess glucose produced doesn’t seem to follow a very well defined behaviour, but again this is due to too many unknown variables on how the bacteria would handle the situation.

If we had more time we could fiddle around with the enzyme rates, rate constants and other parameters in order to find a better prediction, and maybe get some more relevant data like for example how many bacteria are necessary for sufficient inulin generation or how much dietary sucrose we could remove from the host. What we have shown is that the system works as an inulin generator with inulin of DP=5 which is the main purpose after all.

Source code

METHOD RK4
STARTTIME = 0
STOPTIME=100
DT= 0.02
;Promotor activities
;Heat regulated promoter
PA=DELAY(Kact,5,0) ; at time=5 it goes from hot to cold ie. from zero activity to Kact activity (the last zero is degrees heat, so if the system is in a fridge that is four degrees C then the zero should be 4, Kact should then be body heat 37 degeree C)
;constitutiv promotor
;d/dt(PA)=0
;INIT PA=Kact
;Sucros dependent promoter
PA=Kact*S/(Ksp+S)
;Dependent variables
;S=Sucrose concentration
;G=Glucose concentration
;F=Fructose concentration
;T= trisaccharide concentration
;Inu1=Concentration of inulin 1
;Inu2=Concentration of inulin 2
;Inu3=Concentration of inulin 3
;Inu4=Concentration of inulin 4
;Inu5=Concentration of inulin 5
;Bak=amount of bacteria
;SST=amount of SST enzyme
;FFT=amount of FFT enzyme
;Initial values
INIT S=0.01 ;grams per litre
INIT G=0
INIT F=0
INIT T=0
INIT Inu1=0
INIT Inu2=0
INIT Inu3=0
INIT Inu4=0
INIT Inu5=0
INIT Bak=1; one gram
INIT SST=0
INIT FFT=0
;Rates
;Rxxt=XXT combination of the sucrose catalyzation of both SST and FFT into Fructose and Glucose (follows basic michaelis-menten kinetics)
;Rsst=SST enzyme rate fortrisaccharide production
;Rbac= bacterial growth from surplus glucose
;Rfft0= FFT enzyme rate for inulin1
;Rfft1= FFT enzyme rate for inulin2
;Rfft2= FFT enzyme rate for inulin3
;Rfft3= FFT enzyme rate for inulin4
;Rfft4= FFT enzyme rate for inulin5
;TRsst= translation rate of SST
;TRfft= translation rate of FFT
;EzDs=enzyme decay rate for SST
;EzDf=enzyme decay rate for FFT
;BacD=rate of death for bacteria
;Differential equations
d/dt(Bak)=Rbac-BacD
d/dt(SST)=TRsst-EzDs
d/dt(FFT)=TRfft-EzDf
d/dt(S)=Ksuc-Rxxt ;set Ksuc=0 and INIT S high to simulate massive sugar consumption for host
d/dt(G)=Rxxt-Rbac-Rsst
d/dt(F)=Rxxt-(2*Rsst)-Rfft0-Rfft1-Rfft2-Rfft3-Rfft4
d/dt(T)=Rsst-Rfft0
d/dt(Inu1)=Rfft0-Rfft1
d/dt(Inu2)=Rfft1-Rfft2
d/dt(Inu3)=Rfft2-Rfft3
d/dt(Inu4)=Rfft3-Rfft4
d/dt(Inu5)=Rfft4


;__________________________ ;RATES
TRsst=Ktrsst*PA*Bak
TRfft=Ktrfft*PA*Bak
BacD=Kdeath*Bak
EzDs=Kezd*SST
EzDf=Kezd*FFT
Rxxt=Vxxt*(SST+FFT)*S/(Kxxt+S)
Rsst=Vsst*SST*(G/(Ksst+G))*((F^2/(Kfft2+F^2)))
Rbac=Kbak*G
Rfft0=Vfft*FFT*(T/(Kfft1+T))*(F/(Kfft2+F))
Rfft1=Vfft*FFT*(Inu1/(Kfft1+Inu1))*(F/(Kfft2+F))
Rfft2=Vfft*FFT*(Inu2/(Kfft1+Inu2))*(F/(Kfft2+F))
Rfft3=Vfft*FFT*(Inu3/(Kfft1+Inu3))*(F/(Kfft2+F))
Rfft4=Vfft*FFT*(Inu4/(Kfft1+Inu4))*(F/(Kfft2+F))


;Constants
Vxxt=1
Vsst=1
Vfft=1
Kxxt=1
Ksst=1
Kbak=1
Kfft1=1
Kfft2=1
Ktrsst=1
Ktrfft=1
Kezd=0.01
Ksuc=1
Kact=1
Kdeath=0.1
Ksp=1