EE 5393: Circuits, Computation and Biology

1. AND OR AND Marc D. Riedel Associate Professor, ECE University of Minnesota EE 5393: Circuits, Computation and Biology

2. Playing by the Rules Rules for integrated circuits: amplifier v1 1 0 rin1 1 0 9e12 rjump 1 4 1e-12 rin2 4 0 9e12 e1 3 0 1 2 999k e2 6 0 4 5 999k e3 9 0 8 7 999k rload 9 0 10k r1 2 3 10k rgain 2 5 10k r2 5 6 10k r3 3 7 10k r4 7 9 10k r5 6 8 10k r6 8 0 10k .dc v1 0 10 1 .print dc v(9) .end SPICE circuitnetlist waveforms

3. Playing by the Rules Rules for biochemistry: SPICE Gillespie’sSSA X=100, Y = 30Xa = Xb = Xn= 0Y = 0 histogram: resulting quantities of proteins biochemical reactions and initial quantities of proteins

4. Playing by the Rules Rules for biochemistry: SPICE Gillespie’sSSA algorithms widely studied data structures (Gibson & Bruck, Fett & Riedel); approximation methods (Petzold); hybrid discrete/continuous methods (Kaznessis); …

5. dynamics well studied mathematics (Tyson, Khammash, Doyle, …); biology (Arkin, Endy, Brent); … computation (Winfree, Shapiro); … SPICE Gillespie’sSSA X=100, Y = 30Xa = Xb = Xn= 0Y = 0

6. Biochemical Netlists Netlists found in nature: • Elucidated by biologists. New Netlists: • Designed by skilled experimentalists(by tinkering with existing mechanisms). X=100, Y = 30Xa = Xb = Xn= 0Y = 0 Where does the netlist come from?

7. Playing by the Rules + 2a c b + Biochemical Reactions: how types of molecules combine.

8. Biochemical Reactions + cell species count 9 8 6 5 7 9 Discrete chemical kinetics; spatial homogeneity.

9. Biochemical Reactions + + + Relative rates or (reaction propensities): slow medium fast Discrete chemical kinetics; spatial homogeneity.

10. R1 R2 R3 Stochastic Kinetics The probability that a given reaction is the next to fire is proportional to: • Its rate. • The number of ways that the reactants can combine. See D. Gillespie, “Stochastic Chemical Kinetics”, 2006.

11. Ri Stochastic Kinetics For each reaction let Choose the next reaction according to:

12. Biochemical Reactions Lingua Franca of computational biology. Reaction 1 molecule of type A combines with 2 molecules of typeB to produce 2 molecules of type C. Reaction is annotated with a rate constant and physical constraints (localization, gradients, etc.)

13. Elementary step (e.g., ) Biochemical Reactions Lingua Franca of computational biology. Reaction Species: • Elementary molecules (e.g., hydrogen, phosphorous, ...) • Complex molecules (e.g., proteins, enzymes, RNA ...) Reaction: • Conglomeration of steps (e.g., transcription of gene product)

14. R1 R2 R3 Biochemical Reactions Lingua Franca of computational biology. Coupled Set Reactions Goal: given initial conditions, analyze (predict) the evolution of such a system.

15. Biochemical Reactions Convential Approach: numerical calculations based on coupled ordinary differential equations. • Computationally challenging (sometimes intractable). • Assumes that molecular quantities are continuous values that vary deterministically over time.

16. Biochemical Reactions Convential Approach: numerical calculations based on coupled ordinary differential equations. • In intracellular networks, the number of molecules of each complex type is generally small (10s, 100s, at most 1000s). • Individual reactions matter.

17. R1 R2 R3 e.g., Gillespie’s Framework Track precise (integer) quantities of molecular species. “States” Reactions A B C S1 4 7 5 S2 2 6 8 S3 22 0 997 A reaction transforms one state into another:

18. R1 R2 R3 StochasticSimulation S1 = [5, 5, 5] 0 Ri Choose the next reaction according to: where

19. R1 R2 R3 StochasticSimulation S1 = [5, 5, 5] 0 Ri Choose the time of the next reaction according to:

20. R1 R2 R3 StochasticSimulation S1 = [5, 5, 5] 0 See D. Gillespie, “Exact Stochastic Simulation of Coupled Chemical Reactions”,J. Phys. Chem. 1977

21. StochasticSimulation S1 = [5, 5, 5] 0 Choose R3 and t = 3 seconds. R1 R2 R3 S2 = [4, 7, 4] 3 Choose R1 and t = 1 seconds. S3 = [2, 6, 7] 4 Choose R3 and t = 2 seconds. S4 = [1, 8, 6] 6 Choose R2 and t = 1 seconds.

22. StochasticSimulation S1 = [5, 5, 5] 0 Choose R3 and t = 3 seconds. S2 = [4, 7, 4] 3 7 Choose R1 and t = 1 seconds. S3 = [2, 6, 7] 4 Choose R3 and t = 2 seconds. S4 = [1, 8, 6] 6 Choose R2 and t = 1 seconds.

23. A + N B B X B + N C C Y C A + 2N Is Looping Necessary? 1 0.001 2 Reactions 0.002 3 • If we begin with 1 molecule ofAand 2 molecules ofN: • what is the probability that we get a molecule of X? • what is the probability that we get a molecule of Y? (assuming that we wait as long as it takes)

24. A + N B B X B + N C C Y C A + 2N SA SB SC SX SY Is Looping Necessary? 1 0.001 2 Reactions 0.002 3 Trial 1 Trial 2 Cycle 1000 times ... Now repeat 500 times ...