Computational models: Reaction Diffusion. Matthias Vigelius Week 6

Transcription

1 Computational models: Reaction Diffusion Matthias Vigelius Week 6

2 Reactions

3 (Chemical) reactions Reaction is a transformation from reactants into products: A + 2B D + 3E + k k [ ] Reactions occur with a certain rate Elementary reactions are: s 1 (per second). Equation Type Rate A A B A + B C zeroth order first order second order k ka kab

4 Examples Birth reaction: Radioactive decay: Predator-Prey interaction: +

5 Simplistic model of gene expression The central dogma of molecular biology describes the way genetic information is expressed in all organisms: DNA mrna mrna Protein Exercise: Model this simple dogma in Insight Maker! Alberts et al.: Molecular Biology of the Cell (2002)

6 Simplistic model of gene regulation SIMPLISTIC GENE EXPRESSION MODEL I: NO REGULATION Loading Insight Maker... (This may take a few moments)

7 Gene regulation Proteins can switch genes on or off, which allows the cell to adapt to changing environmental conditions. What if a protein regulates its own production (feedback)?

8 Simplistic model of gene regulation GENE EXPRESSION MODEL II: REGULATION Loading Insight Maker... (This may take a few moments)

9 Feedback Feedback is associated with closed loops in the dynamic description (feedback loops) Negative feedback tends to stabilize systems (Exception: lecture feedback) Positive feedback destabilizes. If unchecked, things get out of control. Feedback is a central concept in nature and engineering. Group discussion: Find more feedback examples (positive/negative)!

10 Nuclear fission from: A. Zabludoff, Nuclear reactions in the sun Positive feedback: Neutrons induce fission step Fission produces more neutrons Can get out of control.. We can write the fission step as an autocatalytic reaction: n U 3n + 90 Kr Ba

11 Structure formation in the early universe

12 Diffusion

13 Microscopic model of diffusion Wiki Diffusion is caused by randomly moving entities Movement may be active (animals) or passive (solute in solvent, Brownian motion) Diffusion tends to smooth out concentration differences (gradients) If particles can interact: reaction-diffusion. Many applications in nature!

14 Hierarchy of models Microscopic level Describe motion of individual particles Mesoscopic level Count number of particles in cell Macroscopic level Let cell size become small (concentration!) Adapt the scale of your model to the problem! Question: How many molecules in your pencil?

15 Reaction diffusion: Pattern formation CDF plugin not found! Please download it here. The brusselator applet is here.

16 Reaction-diffusion: Pattern formation Further reading: Turing patterns activator species stimulates its own production (autocatalysis) and the production of the inhibitor inhibitor inhibits production of activator Inhibitor diffuses fast (away from peak) Activator diffuses slowly (stays at peak) This is where the zebra gets its stripes from.

17 Belousov Zhabotinsky (BZ) reaction Source: YouTube

18 Reaction scheme I: FKN model HOBr + Br + H + Br 2 + H 2 O HBr O 2 + Br + H + 2HOBr Br O + Br + 2H + HBr O + HOBr 3 2 2HBrO 2 Br O + HOBr + 3 H + Br O + HBr O + 2Br + O 3 2 H + O 2 H 2 Br O 3 + Ce 3+ + H + HBr O 2 + Ce 4+ Br O 2 + Ce 4+ + H 2 O Br O + Ce H + 3 Br 2 + CH 2 (COOH) 2 BrCH(COOH) 2 + Br + H + 6 Ce 4+ + CH 2 (COOH) 2 + 2H 2 O 6 Ce 3+ + HCOOH + 2 CO 2 + 6H + 4 Ce 4+ + BrCH(COOH) 2 + 2H 2 O Br + 4 Ce 3+ + HCOOH + 2C O 2 + 5H + Richard J. Field, Endre Koros, and Richard M. Noyes (1972): J. Am. Chem. Soc. 94

19 Separation of Time Scales Imagine catalytic reaction with intermediate: S + E X P + E fast Standard example: Enzyme kinetics Keener and Sneyd: Mathematical Physiology I [Question: Why is this hard to simulate?] The intermediate product is always in quasi steady-state equilibrium with the substrate. So we can simplify: S + E P + E slow

20 Reaction Scheme II: Oregonator model Using separation of time scales we can reduce the system to five equations: Scholarpedia: Oregonator A + Y X + P X + Y 2P A + X 2X + 2Z X + X A + P 1 B + Z Y 2 XZ We have three dynamic species:, and. Question: Where are the positive and negative feedback reactions? Y

21 Oregonator model Source: Vigelius/Meyer (2012)