Event-Leaping in the Stochastic Simulation of Biochemistry State Space AnalysisThe Goddess Durga Marc Riedel, EE5393, Univ. of Minnesota

Biochemical Reactions inputsoutputs Quantities of Different Types of Molecules computation View intra-cellular biochemistry as a form of computing. Chemical Reactions 

inputsoutputscomputation A = 1000 B = 333 C = 666 A = 0 B = 1334 C = 226  Chemical Reactions Biochemical Reactions View intra-cellular biochemistry as a form of computing. However, output is often stochastic.

inputsoutputscomputation A = 1000 B = 333 C = 666  Chemical Reactions Biochemical Reactions View intra-cellular biochemistry as a form of computing. C 1 : A 500 C 2 : B > 500 before A < 550 termination conditions Pr(C 1 ) = 0.61 Pr(C 2 ) = 0.39 A = 0 B = 1334 C = 226

Biochemical Reactions inputsoutputs Quantities of Different Types of Molecules Probability Distribution on terminal conditions computation View intra-cellular biochemistry as a form of computing. Chemical Reactions 

Model as a discrete Markov process. “States” ABC S1S1 S2S2 S3S3 A reaction transforms one state into another: e.g., Probabilistic Analysis Reactions R1R1 R2R2 R3R3

Why? Helpful for the purposes of analysis. Suggests the possibility of synthesis. Expertise from ECE can be brought to bear: algorithms, data structures, abstractions... Engineer a form of “logical” control of biochemical processes: outputs that depend on specific combinations of inputs. Biochemical Reactions View intra-cellular biochemistry as a form of computing.

See D. Gillespie, “Exact Stochastic Simulation of Coupled Chemical Reactions”, J. Phys. Chem R1R1 R2R2 R3R3 S 1 = [5, 5, 5] 0 Computing the Probability Distribution

S 1 = [5, 5, 5] 0 S 2 = [4, 7, 4] Choose R 3 and t = 3 seconds. R1R1 R2R2 R3R3 S 3 = [2, 6, 7] 4 Choose R 1 and t = 1 seconds. S 4 = [1, 8, 6] 6 Choose R 3 and t = 2 seconds. 3 Choose R 2 and t = 1 seconds. Computing the Probability Distribution

Randomness Pseudo-random numbers needed: R1R1 R2R2 R3R3 R4R4 probabilities generate a random number: choose R 2

choose R 2 Randomness Pseudo-random numbers needed: R1R1 R2R2 R3R3 R4R4 probabilities 01 generate a random number: choose R 4

Randomness Pseudo-random numbers needed: Generating random numbers is time consuming. If variance in probabilities is large, accuracy is wasted. R1R1 R2R2 R3R3 R4R4 probabilities 01 generate a random number: choose R 4

Event Leaping Explore high probability events further Along each path, probabilities are multiplicative.

Event Leaping Explore high probability events further When paths merge, probabilities are additive Along each path, probabilities are multiplicative.

Event Leaping Based on a single random number, leap directly to the boundary of explored region Explore high probability events further. When paths merge, probabilities are additive.

We need far fewer random numbers (e.g., factor of 10 reduction). We cache probability calculations. If we return to any portion of the region already visited, we immediately jump to the boundary of it. Event Leaping

State Space Analysis Characterize Evolution [0, 0, 12] [1, 1, 9][1, 5, 4][4, 4, 0][4, 0, 5] [2, 2, 6][2, 6, 1][5, 1, 2] p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 p 12 p ppp )()(ppppppppppppp  [3, 3, 3] start [3, 3, 3] e.g., identify articulation points

State Space Analysis Characterize Evolution [2, 2, 6][2, 6, 1][5, 1, 2] [3, 3, 3] start e.g., identify “articulation” points Pr(C 1 ) > 0.99Pr(C 2 ) > 0.99

Synthesis Impose logical structure on computation. AND gate Conjunction of types X and Y : X Y Z N.B. operation is destructive Negation of X : T is a “source” (initialize to ≈ 1000) N is a neutral type X X If X ≈ 1000, T ≈ 0 If X ≈ 0, T ≈ 1000 NOT gate

Probabilistic Analysis Suppose:What is Analyze timing, “leakage currents”, etc. Reaction k1k1

Computing with Probabilistic Gates AND X Y Z p p   1)incorrectPr( )correctPr( q q   1)0X )1X q q   1)0Y )1Y 22 2)0 qpqpZ  22 21)1 qpqpZ 

Research Themes Conceptual Problems in Logic Design: models, physical attributes, complexity. Computational Approach: symbolic data structures, simulation, massively parallel computing. Application of Expertise to Computational Biology OR AND xxabcd