Applications of Differential Equations in Synthetic Biology

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