Team:CLS CLSG UK/Enzyme concentration
Credit - Volkan Olmez@unsplash
Enzyme Concentration
Constants
T1/2 - half life
kt - time constant for thermal decay
kc - time constant for cellular decay
Et - the variable number of enzymes if only subject to thermal decay
Ec - the variable number of enzymes if only subject to cellular decay
Kd - time constant for both thermal and cellular decay
Kp - the rate at which the enzyme is created
An iterative model to to measure maximum enzyme concentration
For our final model we required a value for the constant Vmax which we couldn’t find in the literature. Since Vmax=kcat∙[E]0 and we had a value for kcat we needed to calculate a value for the initial enzyme concentration.
Biologically, the production of the enzyme can be described as:
The rate change of enzyme concentration is therefore equal to the rate of production from mRNA minus the rate of cellular and thermal decay of the enzyme.
The rate of production is known to be a constant value which we can calculate if we know the number of enzymes produced per generation. And the amount of time it takes to produce one generation.
Number of enzymes produced per generation = 1000
Time to produce one generation = 17 minutes
The rate of enzyme production (which we later refer to as Kp) is therefore 1000/1020 = 0.980 s-1
The time it takes for each enzyme to decay thermally is expressed bellow where t1/2 is the half-life of an enzyme decaying purely thermally.
t1/2=12 minutes for the CocE
t1/2=360 minutes for the mutant enzyme
The general equation for exponential decay is:
And therefore when A= 1/2A0, i.e. at time t1/2
kt=ln(2)/720=9.63×10-4s-1 for CocE
kt=ln(2)/21600=3.21×10-5s-1 for the mutant enzyme
The constants can be expressed very similarly for cellular decay where t1/2 is the half-life of an enzyme decaying purely cellularly.
t1/2=900 minutes for both enzymes
kc=ln(2)/54000=1.28×10-5s-1
We can write two general exponential expressions for the concentration of enzyme subject to both thermal and cellular decay separately as so:
Differentiating both gives two rates which can be added linearly as so:
Integrating equation 2.4 this with respect to t gives:
This equation measures the reduction in the number of enzymes, ΔE as a function of time. However, more enzyme is being constantly produced and therefore E0 is constantly changing. This resulted in us creating an iterative model on MATLAB where we modelled the increase in enzyme concentration for tiny increments of time and added this increase to the total enzyme concentration from the previous iteration. We crucially imputed the value for the total enzyme concentration from the previous iteration as a value for E0.
Our model used this equation for the change in enzyme concentration δE:
where Kd=-(kc+kt) from equation 2.4 and 2.5 and Kp is the constant from equation 2.1 for the rate of production
We coded the resulting iterative model on MATLAB. The model becomes more accurate as dt decreases and the number of iterations, increases. However this simple program takes longer and longer to run these two values become more extreme. Since the number of enzymes produced each minute can only reliably known to 2 significant figures these values for dt and the number of iterations were sufficient. It gave a resulting value of 1000 (2 s.f.) for CocE and 22000 (2 s.f.) for the mutant enzyme
The results can be shown graphically (graph 2 and 3), as we then did on Microsoft Excel. The shape of the graph matches predictions that the rate of change in the number of enzymes should decrease exponentially.