User:Jaroslaw Karcz/Sandbox

20 September 2007

It was suggested that the levelling-off of the fluorescence-time curve for construct 1 (T9002) [1] could more realistically be attributed to the extinction of "energy" in the system. The model for Infector Dectector needed to be amended, to include this phenomenon.

Microbial growth rate as function of single rate-limiting substrate

Although a study of microbial growth rate is undertaken here, this could be altered to feature the system survival/lifetime, as we are dealing with S30 cell extract. The lifetime of the system would be reflected by the rate of change of system energy ([math]\displaystyle{ {d[nutrient]}/dt }[/math]), which here is decreasing, as there is no source of replenishment of nutrient.

A review of literature suggests that multiple models have been developed to describe this feature of the system. The most widely used models are the Monod, Grau, Teisser, Moser and Contois equations.
These equations describe the functional relationship between the microbial growth rate and essential substrate (nutrient) concentration.

It was proposed, and noted from experimental data, that the behaviour of the system can be described as being limited by the energy of the system, and this could be achieved by using the Hill function.

[math]\displaystyle{ \mu_{obs} = \mu_{max}\frac{[S]^n}{K_s^n + [S]^n} \cdots (1) }[/math]

where

• S = limiting nutrient/substrate ("energy in system")
• [math]\displaystyle{ \mu }[/math] = instantaneous (observed) growth rate coefficient
• [math]\displaystyle{ \mu_{max} }[/math] = maximal growth rate coefficient
• K_s = half-saturation coefficient
• n = positive co-operativity coefficient
  
  

For Infector Detector, let [math]\displaystyle{ S \rightarrow E }[/math] such that (1) becomes

[math]\displaystyle{ \mu = \mu_{max}\frac{[E]^n}{K_s^n + [E]^n} \cdots (2) }[/math]

where E represents the energy of the system, and [math]\displaystyle{ \mu }[/math], effectively, the efficiency of the system. Also, take [math]\displaystyle{ \mu_{max} }[/math] equal to 1, such that maximal system efficiency is attained at [math]\displaystyle{ \mu }[/math] = 1. K_s = K_E; the half-saturation coefficient.

Furthermore, the energy term, E, needs to be included in the biochemical network equations of our system as follows:

[math]\displaystyle{ \mu = \frac{[E]^n}{K_E^n + [E]^n} }[/math]

[math]\displaystyle{ \frac{d[LuxR]}{dt} = k_1\mu + k_3[A] - k2[LuxR][AHL]- \delta_{LuxR}[LuxR] }[/math]
[math]\displaystyle{ \frac{d[AHL]}{dt} = k_3[A] - k2[LuxR][AHL]- \delta_{AHL}[AHL] }[/math]

[math]\displaystyle{ \frac{d[A]}{dt} = -k_3[A] + k2[LuxR][AHL]- k_4[A][pLux] + k_5[AP] }[/math]

[math]\displaystyle{ \frac{d[AP]}{dt} = k_4[A][pLux] - k_5[AP] - k_6[AP] }[/math]

[math]\displaystyle{ \frac{d[GFP]}{dt} = k_6[AP]U - \delta_{GFP}[GFP] }[/math]

[math]\displaystyle{ \frac{d[E]}{dt} = -k_1\mu - k_6[AP]\mu }[/math]

Koch and Schaechter studied the effect of glucose concentration ([E]) on the observed growth rate of E. coli in a pure culture. The value for the co-operativity coefficient, n, was found to 2.38.