# 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 $\dot{m}=k_{to} - d_{to}m$ $\dot{p_h} = k_hm - d_hp_h$ $\dot{p_t} = k_tp_h - d_tp_t$ $\dot{p_g} = k_gp_t - d_gp_g$ Parameters kto...transcription rate of HIV1 dto...degradation rate of mRNA coding for HIV1 kh...translation rate of HIV1 dh...degradation rate of HIV1 kt...production rate of TEV by HIV1 dt...degradation rate of TEV kg...production rate of GFP by TEV dg...degradation rate of GFP

### TEV

 TEV is used at both stages of amplification Equations $\dot{m} = k_{to} - d_{to}m$ $\dot{p_t} = k_tm - d_tp_t$ $\dot{p_{ts}} = k_{ts}p_t - d_{ts}p_{ts}$ $\dot{p_g} = k_{g1}p_t + k_{g2}p_{ts} - d_gp_g$ Parameters kto...rate of transcription by TEV dto...degradation rate of mRNA coding for TEV kt...rate of translation of TEV dt...degradation rate of TEV kts...rate of production (fusion) of split TEV dts...degradation rate of split TEV kg1...rate of production of GFP by full TEV kg2...rate of production of GFP by split TEV dg...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 $\dot{p_t} = s_t - d_tp_t$ $s_t = \dfrac{k_tk_{to}}{d_{to}}$ 2. Production of split TEV from transcription $\dot{p_{st}} = s_{st} - d_{st}p_{st}$ 3. Production of split GFP from transcription $\dot{p_{sg}} = s_{sg} - d_{sg}p_{sg}$ 4. Production of fused split TEV catalysed by TEV (1) $\dot{p_{ts}} = \dfrac{V_{max,t}[p_{st}]}{K_{m,ts} + [p_{st}]} - d_{ts}p_{ts}$ 5. Production of GFP catalysed by TEV (1) and fused split TEV (4) $\dot{p_g} = \dfrac{V_{max,tg}[p_{sg}]}{K_{m,tg} + [p_{sg}]} + \dfrac{V_{max,tsg}[p_{sg}]}{K_{m,tsg} + [p_{sg}]} - d_gp_g$

### 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]