1. Applications of Differential Equations in Synthetic Biology
John Sy

2. What is a differential equation?
A differential equation is a mathematical expression involving the derivatives of variables – a strict definition
They are used to model real life situations
Uses?
Physics – Schrodinger, atomic physics
Chemistry – Rate Laws, Statistical Thermo
…Synthetic Biology (?)

3. Example 1: Radioactive Half-Life
A stochastic (random) process
The RATE of decay is dependent upon the number of molecules/atoms that are there
Negative because the number is decreasing
K is the constant of proportionality

4. Example 2: Rate Laws
An integrated rate law is an expression involving the rate of reaction and the species involved in the reaction
Consider the reaction:
The rate of change of A is dependent upon the concentration of A

5. Rate Laws We can model this by the differential equation:
Note: -ve sign and k constant

6. First Order Rate Laws
First Order since it only depends upon the concentration linearly

7. Second & Higher Order Rate Laws
A Second Order rate law would depend upon the square of the concentration:
An nth order rate law would depend upon the nth power of the concentration:

8. Connecting to Biological Reactions
Consider transcription, the rate is constant and is not dependent upon the concentration of any of the species involved
Although the transcription rate does depend on several other factors, we can formulate a simple model which assumes that the transcription rate is constant
[mRNA] vs. time (x-axis)

9. Connecting to Biological Reactions
Consider degradation of either mRNA or protein…

10. Connecting to Biological Reactions
Now consider the phenomenon of translation
The rate of translation is dependent upon the amount of mRNA
k is the rate constant to be determined experimentally

11. Combining Everything Together
Now consider a protein synthesis reaction where the mRNA and protein are degraded:
The first term is the transcription rate and the second term is the degradation rate

12. An equilibrium is reached such that the rate of translation balances out the rate of degradation
conc of protein
Is this what we expect and why?
conc of mRNA

13. Michaelis-Menten Kinetics
Consider a biological system with an enzyme
If we consider the rates of reactions of all three reactions going on, then we can derive the rate equation (note we are considering the concentration of the Product):

14. Michaelis-Menten Consequences
The relationship yields the hyperbolic relationship between the velocity (rate of product production) and the substrate concentration
NB that this equation is not directly dependent upon the enzyme concentration
It is hidden within the maximum velocity

15. Activators
An activator works similar to an enzyme, it can speed up the transcription rate, so we can model activators by Michaelis-Menten kinetics
NB as the rate of reaction is not dependent upon enzyme concentration directly, transcription rate is also not dependent upon activator concentration directly

16. Repressors
Repressors are slightly different…the absence of the repressor will determine the rate of transcription, so we need to modify the rate equation from the activator:

17. How do we solve these equations?
Depends if equations are linear or non-linear and what order it is
First order & second order linear – separation of variables, standard techniques
Higher order and non-linear – integral transforms and approximation methods
…otherwise, you can always use a computer (mathematica, maple or matlab)

18. CellDesigner
The CellDesigner tutorial will help you to visualize these graphs and to familiarize yourself with the maths behind this project!
A powerful tool that allows you to:
Design a block model of the system
Implement your kinetic law
Visualize the results