IGEM:Imperial/2010/Michaelis Menten

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.

1. How beneficial is the use of amplification? (Compare speed of response of transcription (and translation) with 1- or 2-step amplification)
2. How many amplification steps are beneficial to have? Will further adding of amplification steps introduce too many time delays?
3. 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]\displaystyle{ \dot{m}=k_{to} - d_{to}m }[/math] [math]\displaystyle{ \dot{p_h} = k_hm - d_hp_h }[/math] [math]\displaystyle{ \dot{p_t} = k_tp_h - d_tp_t }[/math] [math]\displaystyle{ \dot{p_g} = k_gp_t - d_gp_g }[/math]

Parameters k_to...transcription rate of HIV1 d_to...degradation rate ogf mRNA coding for HIV1 k_h...translation rate of HIV1 d_h...degradation rate of HIV1 k_t...production rate of TEV by HIV1 d_t...degradation rate of TEV k_g...production rate of GFP by TEB d_g...degradation rate of GFP

TEV

TEV is used at both stages of amplification

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

Parameters k_to...rate of transcription by TEV d_to...degradation rate of mRNA coding for TEV k_t...rate of translation of TEV d_t...degradation rate of TEV k_ts...rate of production (fusion) of split TEV d_ts...degradation rate of split TEV k_g1...rate of production of GFP by full TEV k_g2...rate of production of GFP by split TEV d_g...degradation rate of GFP

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 p_t' = s_t - d_t * p_t; where s_t = (k_t * k_to)/d_to 2. Production of split TEV from transcription p_st' = s_st - d_st * p_st 3. Production of split GFP from transcription p_sg' = s_sg - d_sg * p_sg 4. Production of fused split TEV catalysed by TEV (1) p_ts' = (V_max,t * [p_st])/(K_m,ts + [p_st]) - d_ts * p_ts 5. Production of GFP catalysed by TEV (1) and fused split TEV (4) p_g' = (V_max,tg * [p_sg])/(K_m,tg + [p_sg]) + (V_max,tsg * [p_sg])/(K_m,tsg + [p_sg]) - d_g * p_g

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
Km and Kcat	-	Km = 0.061; Kcat = 0.16; [1]	40% of value for TEV	-
Half-life or degradation rate	Half-life in B.sub approximately 1.5 hours	?	?	Half-life shorter than GFP
Production rate in B.sub	?	?	?	?

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

1. Kapust R. et al (2001) Tobacco etch virus protease: mechanism of autolysis and rational design of stable mutants with wild-type catalytic proficiency. Protein Engineering. [Online] 14(12), 993-1000. Available from: http://peds.oxfordjournals.org/cgi/reprint/14/12/993 [Accessed 28th July 2010]