Model based on Michaelis Menten Kinetics (Weeks 4 and 5)
Motivation
We came up with a simple concept of output amplification, which is enhanced by using enzymes. It is beneficial for us to model the behaviour of our design so that we will be able to answer the following questions.
 How beneficial is the use of amplification? (Compare speed of response of transcription (and translation) with 1 or 2step amplification)
 How many amplification steps are beneficial to have? Will further adding of amplification steps introduce too many time delays?
 Is it better to use TEV all or HIV1?
Modelling should allows us to make a decision on which design is the most efficient one.
First Model
HIV1
At each stage of amplification a distinct protease is being used

Equations
 [math]\dot{m}=k_{to}  d_{to}m[/math]
 [math]\dot{p_h} = k_hm  d_hp_h[/math]
 [math]\dot{p_t} = k_tp_h  d_tp_t[/math]
 [math]\dot{p_g} = k_gp_t  d_gp_g[/math]

Parameters
 [math]k_{to}\mbox{...transcription rate of HIV1}[/math]
 [math]d_{to}\mbox{...degradation rate of mRNA coding for HIV1}[/math]
 [math]k_h\mbox{...translation rate of HIV1}[/math]
 [math]d_h\mbox{...degradation rate of HIV1}[/math]
 [math]k_t\mbox{...production rate of TEV by HIV1}[/math]
 [math]d_t\mbox{...degradation rate of TEV}[/math]
 [math]k_g\mbox{...production rate of GFP by TEV}[/math]
 [math]d_g\mbox{...degradation rate of GFP}[/math]

TEV
TEV is used at both stages of amplification

Equations
 [math]\dot{m} = k_{to}  d_{to}m[/math]
 [math]\dot{p_t} = k_tm  d_tp_t[/math]
 [math]\dot{p_{ts}} = k_{ts}p_t  d_{ts}p_{ts}[/math]
 [math]\dot{p_g} = k_{g1}p_t + k_{g2}p_{ts}  d_gp_g[/math]

Parameters
 [math]k_{to}\mbox{...rate of transcription by TEV}[/math]
 [math]d_{to}\mbox{...degradation rate of mRNA coding for TEV}[/math]
 [math]k_t\mbox{...rate of translation of TEV}[/math]
 [math]d_t\mbox{...degradation rate of TEV}[/math]
 [math]k_{ts}\mbox{...rate of production (fusion) of split TEV}[/math]
 [math]d_{ts}\mbox{...degradation rate of split TEV}[/math]
 [math]k_{g1}\mbox{...rate of production of GFP by full TEV}[/math]
 [math]k_{g2}\mbox{...rate of production of GFP by split TEV}[/math]
 [math]d_g\mbox{...degradation rate of GFP}[/math]

Improved Model which accounts for enzyme reactions (28/07/2010)
TEV
TEV is used at both stages of amplification

Equations
 1. Production of TEV from transcription
[math]\dot{p_t} = s_t  d_tp_t[/math]
[math]s_t = \dfrac{k_tk_{to}}{d_{to}}[/math]
 2. Production of split TEV from transcription
[math]\dot{p_{st}} = s_{st}  d_{st}p_{st}[/math]
 3. Production of split GFP from transcription
[math]\dot{p_{sg}} = s_{sg}  d_{sg}p_{sg}[/math]
 4. Production of fused split TEV catalysed by TEV (1)
[math]\dot{p_{ts}} = \dfrac{V_{max,t}[p_{st}]}{K_{m,ts} + [p_{st}]}  d_{ts}p_{ts}[/math]
 5. Production of GFP catalysed by TEV (1) and fused split TEV (4)
[math]\dot{p_g} = \dfrac{V_{max,tg}[p_{sg}]}{K_{m,tg} + [p_{sg}]} + \dfrac{V_{max,tsg}[p_{sg}]}{K_{m,tsg} + [p_{sg}]}  d_gp_g[/math]

Implementation in Matlab
The Matlab code for the different stages of amplification and diagrams can be found here.
Kinetic constants

GFP

TEV

split TEV

split GFP

[math]Km[/math] and [math]k_{cat}[/math]



[math]K_m = 0.061[/math]; [math]k_{cat} = 0.16[/math]; [1]

40% of value for TEV



Halflife or degradation rate

Halflife in B.sub approximately 1.5 hours

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Halflife shorter than GFP

Production rate in B.sub

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Conclusion
We were not able to obtain all the necessary constants. Hence, we decided to make educated guesses about possible relative values between the constants as well as varying them and observing the change in output.
As the result, we concluded that the amplification happens at each amplification level proposed. The magnitude of amplification varies depending on the constants. There is not much difference between using TEV or HIV1.
References
 Kapust R. et al (2001) Tobacco etch virus protease: mechanism of autolysis and rational design of stable mutants with wildtype catalytic proficiency. Protein Engineering. [Online] 14(12), 9931000. Available from: http://peds.oxfordjournals.org/cgi/reprint/14/12/993 [Accessed 28th July 2010]