Or you just put something like this int_0^1 sum_(i=1)^n i^3=((n(n+1))/2)^2 dx in this website: http://asciimath.org/ RESULT:
In order to assess the real world viability of the BeeT we evaluated the proposed system of application by making a model of the entire system. To do this we used Flux Balance Analysis (FBA) to make model the base chassis. The
chassis The chassis is the base organism that is modified
of BeeT is a variant of Escherichia coli, for which it is known that it does not grow in sugar water, mainly due to high osmotic pressure. 1 The question remained: Does it survive there, and if so, for how long?
What is Flux Balance Analysis
Flux balance analysis (FBA) is a mathematical method for simulating metabolism in genome-scale reconstructions of metabolic networks.
The main components of FBA are reactions and metabolites, which are derived from knowledge about genes and the enzymes they produce to facilitate the reactions between metabolites.
Reactions are represented as rows in a matrix of numbers showing a positive number for production and a negative number for consumption, the numbers representing stoichiometrically balanced chemical equations.
Fluxes are represented as a vector of numbers representing the speed and direction of each reaction. The optimize function will find a parsimonious solution, given a certain objective or multiple objective reaction fluxes to maximize, or minimize if that is the case.
When using FBA the objective is usually the "Biomass reaction", which is a kind of resource sink that consumes all kinds of metabolites to represent growth and cell division, which makes sense given that most cells in the exponential phase will allocate as many resources as possible to cell division.
An important set of reactions is the exchange reactions which represent whether the growth medium itself is either losing or gaining certain metabolites at a certain rate. This represents compounds accumulating in or getting taken up from the medium.
There is also an ATP Maintenance reaction, another resource sink, which represents all ATP required to maintain gene-regulation and regular functioning of a non-growing cell.
Given the published version of the model, 3.15 mmol*gram Dry Weight^-1*hour^-1 was the parsimonious response when maximizing for biomass production whilst minimizing all other functions.
Every flux has certain bounds based on the medium the cell is in and what is known to be biologically possible for a specific organism to take up. These bounds have to be either experimentally determined, but in other cases are put at -1000 for the lower, and +1000 for the upper. The entire range of objective solutions can also be looked for, for every flux, this is known as Flux Variance Analysis, but in general a solution is preferred such that all other fluxes are as low as possible.
"Parsimonious Flux Balance Analysis” was born from the argument that less flux through a certain reaction means less effort the cell has to expend on the production of a certain protein and thereby will be more evolutionarily beneficial in the long term. It is to me the most sensible way to pick a specific solution from the allowed solution space created from a certain Metabolic Linear Programming Problem.
On the BIGG database we found a pre-existing well characterized metabolic model, named iJO1366 from the paper: “A comprehensive genome-scale reconstruction of Escherichia coli metabolism”. Orth JD1, Conrad TM, Na J, Lerman JA, Nam H, Feist AM and Palsson BØ.
In order to work with the model in ssbm format the python package CobraPy was downloaded to facilitate the manipulation and perturbation of the “in silico organism“. [ COBRApy: COnstraints-Based Reconstruction and Analysis for Python ]
The previously mentioned exchange reactions were modified by reading in the reaction ids and their upper and lower bounds. [Link to small page about file.]
Then we changed the objective from Biomass to ATP maintenance in order to represent the ‘starvation mode’ which we assume the cell to be in.
And finally we loop through a range of water efflux rates from the cell and produce a value for the objective, given minimal other fluxes.
Figure 1: The bee-mite-life-cycle (figure 1) shows the integration of these models as highlighted parts. This includes modelling the adaptation of the quorum sensing system (regulation), our optogenetic killswitch (biocontainment), a metabolic model of BeeT (regulation) and the BEEHAVE model (specificity). (CLICK THE GREY AREA TO GO TO THE CONCLUSION)
Key Results
The relationship between max ATP available for survival and water efflux is shown in Figure 1, it demonstrates that there is a linear relation. This implies that if no water is available for ATP used for maintenance outside cell growth, the cell will die. When the model is run without any modification, ie in an environment where it is in the exponential growth phase an ATP Maintenance flux of 3.15 mmol*gram Dry Weight^-1*hour^-1 is given as output by the model.
We do not know the amount needed in sugar water conditions, but because of these results we can start looking at the relationship between survival time and water efflux.
In Figure 2 we can see not only the relationship of survival time against max ATP available for survival, but also how different thresholds of minimal cell-water tolerance would affect this relationship. The minimal cell-water tolerance threshold gives the value at which percentage of the remaining cell-water the point of no return for the cell had been reached. Which has a drastic effect on survival time, changing 20 minutes of maximum survival time to a mere ~90 seconds in the worst case scenario.
Conclusion
Figure 1 shows us that osmotic pressure alone can indeed have an effect on cell regulation and cell death and from Figure 2 it appears that the minimal water allowance threshold has a high impact on range of possible times. We also must accept that the range outside of 90 seconds to 90 minutes is completely undocumented territory as we can only say something about non-infinite values. Because we don't exactly know how much mmol*gDW-1*hour-1 is needed for proper maintenance under harsh conditions, we can not say anything about where on the scale that would be.
What we can say is that if the cells can survive for longer outside of this period, then they must have enough ATP available for basic maintenance, and that if cell death occurs then, that other processes than pure water-efflux must be the cause of that. Perhaps combinations of lack of nutrients and water-efflux, or over production of osmolytes to keep the balance.
References
1. Cheng, Y. L., Hwang, J., & Liu, L. (2011). The Effect of Sucrose-induced Osmotic Stress on the Intracellular Level of cAMP in Escherichia coli using Lac Operon as an Indicator. Journal of Experimental Microbiology and Immunology (JEMI) Vol, 15, 15-21. ↩
