Team:DTU Denmark/modelling

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Modelling of GFP expression


Introduction

Here a model for the GFP expression under the control of “The Redoxilator” expression system is derived. The model links the level of the signal molecules NADH and NAD to the final expression of GFP through the events of transcription factor activation, promoter activation, synthesis and degradation of mRNA and finally synthesis, dilution and degradation of GFP.

NADH and NAD binding to the transcription factor

Both NADH and NAD binds reversible to the same site of the transcription factor (TF):

Dissociation of TF and NADH
Picture 2.png
at equilibrium:
Picture 3.png
Dissociation of TF and NAD:
Picture 4.png
Dissociation of TF and NAD:
Picture 5.png
Balance of N and H. Constant total level of NADH + NAD in the cell are assumed:
Picture 6.png

Balance of TF. TF is constitutively expressed leading to constant total value Picture 7.png.
Picture 8.png

N inhibits DNA binding activity of TF and thus transcription activity. Active TF is designated TF’.
Picture 9.png

Leading to the conserved moiety:
Picture 10.png

Using (3), the fraction of TF’ (tf) can be written as
Picture 11.png

With (1) and (2) in (4) the tf becomes
Picture 12.png

Leading to
Picture 13.png

TFratio.png
Figure 1 - Transcription factor activity with increasing NAD/NADH ratio is hyperbolic. At NAD/NADH = 0.1, tf = 0.5, i.e. half of the transcription factors are active. (For this plot the following parameters were used: K_H = 2 μM, assumed, no experimental value available. K_N = 0,02 μM, NADH affinity > NAD affinity [2]. Picture 7.png = 2mM, estimated TF concentration expressed from a constitutive TEF promoter.)

Promoter binding of active transcription factor

When the active TF (TF') is bound to the promoter P, P becomes active (P’) and transcription is initiated. The number of TF needed for promoter activation is designated m:
Picture 14.png

Here it assumed that binding of TF and TFN is equal.
The dissociation equilibrium for this reaction:
Picture 15.png

The number of promoters is constant, and hence the following sum of fractions is true
Picture 16.png

Where the fraction of active promoter Pf is
Picture 17.png

Introducing P’ from (6) gives
Picture 18.png

Now introducing Picture 37.png we can obtain a function of the fraction of active promoter sites as a function of NADH and NAD level:
Picture 19.png

(For simplicity the expression of (5) has not been inserted).

Pfratio.png
Figure 2 - Fraction of active promoters with increasing NAD/NADH ratio. The curve is sigmoidal and has almost switch like behavior. From NAD/NADH = 0 to 0.01 the difference in active promoter is almost 60 %.(Assumed value of K_P = 1 mM).

mRNA production

mRNA is produced from the transcription of the active promoter. Concentration of intracellular levels of mRNA (Picture 20.png) can be derived from the mRNA balance:
Picture 21.png

Here it is assumed that mRNA is degraded corresponding to a first order reaction and that mRNA is not exported from the cell.
The term Picture 22.png represents the dilution due to growth. However mRNA is highly unstable and is thus degraded within tens of seconds to order few minutes, making k_dm very small compared to growth rate. This makes Picture 24.png, simplifying the mRNA balance to:
Picture 25.png

Production rate of mRNA Picture 26.png
It is assumed that when RNA polymerase is efficiently bound to the promoter, the promoter is active and mRNA is transcribed at constant maximum rate Picture 27.png. The binding of RNA polymerase is dependent on the presence of transcription factor and determined by signal strength (NAD/NADH) as well as Kp. In the overall production rate of mRNA, of course also the number of promoters (and genes) Np is influencing:
Picture 28.png

For a given organism Picture 27.png and Np are constant. Inserting (10) into (9) the mRNA balance becomes:
Picture 35.png

Steady state concentration of mRNA
At steady state (11) is equal to zero since no mRNA accumulates. Here the mRNA concentration is:
Picture 29.png

MRNAratio.png
Figure 3 - mRNA levels at steady state with increasing NAD/NADH ratios. Dilution is neglected since degradation rate of mRNA is much larger. (Assumed parameters: kdm = 8.3 h-1 (half life = 5 min), Np = 60 (2 micron plasmid is used). and Picture 27.png = 180 U/h (assuming 75 bases per second, and 1500 bases in gene).)

GFP production

Finally, the mRNA is used as template for the process of translation to protein, here GFP. The GFP protein balance:
Picture 20.png

As with mRNA it is assumed that GFP is degraded by first order rate. The dilution cannot be neglected since protein degradation is much slower than mRNA degradation. Furthermore it is assumed that GFP is not transported out of the cell.

The protein is synthesized by ribosomes. The number of ribosomes bound to one mRNA (Np) and the protein production rate of the ribosome (kp) should be taken into account. The number of ribosomes in the cell, ATP and available charged amino acids are assumed to be non-limiting. Taking this into account the protein synthesis rate can be described as
Picture 31.png

Making the final GFP balance
Picture 32.png

Looking back, we now have the GFP balance only depending on the levels of NADH and NAD (through mRNA level, promoter activity and transcription factor activity).

Steady state concentration of GFP
From (15) we find that the steady state concentration of GFP is
Picture 33.png

GFPssratio.png
Figure 4 - Steady state GFP levels as a direct function of NAD/NADH signal levels. This model tells the metabolic state (through NAD/NADH ratio) of a cell in steady state. (Assumed parameters Nrib = 1. kp = 288 U/h, assuming 40 amino acids per second. k_dGFP = 1.386 h-1 equal to a half time of 30 min [3]. Growth rate = 0.1 h-1).

The dynamic state

When the cells are exposed to a certain NAD/NADH level mRNA and in turn GFP levels will change in time according to their balances. In figure XX the NAD/NADH ratio is changed from XX (resting state) to a more stressed state where NAD/NADH = XX.
Response.png
Figure 5 - Dynamics of GFP expression when NAD+ level is changed
A) Shows the level of NAD+. From t = 1 to t = 4 the NAD+ = total NAD(H) concentration. B) dynamic mRNA level. A fast response is observed in both production and degradation. C) dynamic GFP levels. The response is much slower than mRNA. After app. 1/2 hour from signal burst, the GFP level has reached half of the maximum level. Degradation and dilution is relatively fast for a protein, the response time is about 40 min here.


If the NAD/NADH level changes in time all values depending on NADH and NAD will also be time depending. In the yeast metabolic cycle NAD/NADH ratio oscillates in time, resulting in oscillation-production of the protein under control of the redox system:
Dynamic2.png
Figure 5 - Dynamics of GFP expression when NAD+ level is changed
A Shows the level of NAD+. From t = 1 to t = 4 the NAD+ = total NAD(H) concentration. B dynamic mRNA level. A fast response is observed in both production and degradation. C dynamic GFP levels. The response is much slower than mRNA. After app. 1/2 hour from signal burst, the GFP level has reached half of the maximum level. Degradation and dilution is relatively fast for a protein, the response time is about 40 min here.


References

[1] Heijnen J.J 2009. Modular structured kinetic models: protein production from genes.
[2] Mark and Paget, 2003. A novel sensor of NADH/NAD….
[3] Mateus and Avery, 2000. Destabilized green fluorescent protein for monitoring dynamic changes in yeast gene expression with flow cytometry. Yeast 2000; 16: 1313±1323.
Modelling facts

Abbreviations used in the model
TF = the transcription factor based on the REX protein
TF' = active transcription factor
N = concentration of NAD
H = concentration of NADH
TFH = concentration of the transcription factor bound to NADH.
TFN = concentration of the transcription factor bound to NAD.
P = Promoter upstream of GFP
P' = activated/transcribed promoter
tf = fraction of active transcription factor
Pf = fraction of active promoter

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