Marc Riedel Synthesizing Stochasticity in Biochemical Systems Electrical & Computer Engineering Jehoshua (Shuki) Bruck Caltech joint work with Brian Fett Univ. of Minnesota CIRCUITS & BIOLOGY RIEDEL UMN University of Minnesota

2 Engineering novel functionality in biological systems. Biochemical Reactions View engineered biochemistry as a form of computation. Synthetic Biology E. Coli computationinputsoutputs Molecular Triggers Molecular Products

3 View engineered biochemistry as a form of computation.Bacteria are engineered to produce an anti-cancer drug: E. Coli Design Scenario drug triggering compound

4 Bacteria invade the cancerous tissue: cancerous tissue Design Scenario

5 cancerous tissue The trigger elicits the bacteria to produce the drug: Design Scenario Bacteria invade the cancerous tissue:

6 cancerous tissue Problem: patient receives too high of a dose of the drug. Design Scenario The trigger elicits the bacteria produce the drug:

7 Design Scenario Bacteria are all identical. Population density is fixed. Exposure to trigger is uniform. Constraints: Control production of drug. Requirement: Conceptual design problem.

8 cancerous tissue Approach: elicit a fractional response. Design Scenario

9 produce drug triggering compound E. Coli Approach: engineer a probabilistic response in each bacterium. with Prob. 0.3 don’t produce drug with Prob. 0.7 Synthesizing Stochasticity

10 Generalization: engineer a probability distribution on logical combinations of different outcomes. cell A with Prob. 0.3 B with Prob. 0.2 C with Prob. 0.5 Synthesizing Stochasticity

11 Generalization: engineer a probability distribution on logical combinations of different outcomes. cell A and B with Prob. 0.3 Synthesizing Stochasticity B and C with Prob. 0.7 A with Prob. 0.3 B with Prob. 0.2 C with Prob. 0.5

12 Generalization: engineer a probability distribution on logical combinations of different outcomes. cell A and B with Prob. 0.3 Synthesizing Stochasticity B and C with Prob. 0.7 Further: program probability distribution with (relative) quantity of input compounds. X Y

13 CAD Engineers doing Biology Why? Specific computational expertise: Cast problems in a computational language: with data structures and algorithms for analyzing and manipulating discrete designs over a large state space. with well-defined, quantitative inputs and outputs; tackling analysis and synthesis systematically. How?

14 Biochemical Reactions 1 molecule of type A combines with 2 molecules of type B to produce 2 molecules of type C. ( specifies the relative rate of occurrence) k Reaction

15 Biochemical Reactions Large types (e.g. proteins, enzymes, RNA). Small quantities (e.g., ~10 3 molecules/cell). Complex interactions. Reaction 1 molecule of type A combines with 2 molecules of type B to produce 2 molecules of type C. ( specifies the relative rate of occurrence) k

16 Discrete Analysis “States” ABC S1S1 S2S2 S3S3 A reaction transforms one state into another: e.g., R1R1 R2R2 R3R3 Track discrete (i.e., integer) quantities of molecular types.

17 S 1 = [5, 5, 5] S 2 = [4, 7, 4] R1R1 R2R2 R3R3 S 3 = [2, 6, 7] S 4 = [1, 8, 6] Discrete Analysis State [ A, B, C ]

18 Discrete Analysis Biochemical Reactions computationinputsoutputs Quantities of Different Types

19 Biochemical Reactions computationinputsoutputs A = 1000 B = 333 C = 666 A = 0 B = 1334 C = 226 Quantities of Different Types Discrete Analysis

20 Probabilistic Analysis The probability that a given reaction is the next to fire is proportional to: Its rate constant (i.e., its k i ). The quantities of its reactants. R1R1 R2R2 R3R3 See D. Gillespie, “Stochastic Chemical Kinetics”, 2006.

21 Probabilistic Analysis Choose the next reaction according to: RiRi let For each reaction

Probabilistic Lattice [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]

23 Biochemical Reactions computationinputsoutputs Quantities of Different Types Probability Distribution on Quantities of Different Types Probabilistic Response

24 X = 30 Y = 40 Z = 30 A with Prob. 0.3 B with Prob. 0.4 C with Prob. 0.3 cell computationinputsoutputs Probabilistic Response Quantities of Different Types Probability Distribution on Quantities of Different Types Biochemical Reactions Found in nature? Achievable by design? Yes.

25 Natural Stochasticity Dead Cell Hijack (Lysis) Stealth (Lysogeny) “Choice” Lambda Bacteriophage (Adam Arkin, 1998)

26 Prob. 0.2 Prob. 0.8 “Portfolio” of Responses Natural Stochasticity Dead Cell “Choice”

27 Synthesizing Stochasticity Contribution of this work: General method for synthesizing a set biochemical reactions that produces a specified probability distribution. Method is: Precise. Robust. Programmable. Modular and extensible.

28 Synthesizing Stochasticity For types d 1, d 2, and d 3, program the response: Example Solution Setup initializing reactions: Initialize e 1, e 2, and e 3, in the ratio: 30 : 40 : 30

29 Setup reinforcing reactions: Synthesizing Stochasticity For types d 1, d 2, and d 3, program the response: Example Solution (cont.)

30 Setup stabilizing reactions: For types d 1, d 2, and d 3, program the response: Example Solution (cont.) Synthesizing Stochasticity

31 Synthesizing Stochasticity Setup purifying reactions: Example Solution (cont.) For types d 1, d 2, and d 3, program the response:

32 Result Synthesizing Stochasticity d1d1 with Prob. d2d2 d3d3 Mutually exclusive production of d 1, d 2, and d 3 : Initialize e 1, e 2, and e 3 in the ratio: x : y : z

33 Initializing Reactions Reinforcing Reactions Stabilizing Purifying Working Reactions where General Method

34 Initializing Reactions Reinforcing Reactions Stabilizing Purifying Working Reactions where General Method

35 Initializing Reactions General Method For all i, to obtain d i with probability p i, select E 1, E 2,…, E n according to: Use as appropriate in working reactions: (where E i is quantity of e i )

36 Error Analysis Let for three reactions (i.e., i, j = 1,2,3). Require Performed 100,000 trials of Monte Carlo. '''''''''' ij iii kkkkk   2 '''''''''',,1  ij iii kkkkk

37 Generalization: engineer a probability distribution with a functional dependence on input quantities. Functional Dependencies cell X Y Approach: deterministic “pre-processing”.

38 Modular Synthesis Deterministic Module... Stochastic Module initializing, reinforcing, stabilizing, purifying, and working reactions linear, exponentiation, logarithm, raising-to-a-power, etc.

39 Synthesizing Stochasticity (potential) Applications: biochemical sensing, drug production, disease treatment. (immediate) Impetus: framework for analyzing and characterizing the stochastic behavior of natural biological systems. Synthesizing Stochasticity in Biochemical Systems

40 Modeling Natural Systems Lambda Bacteriophage (Adam Arkin, 1998) Curve-fits for data from Monte Carlo simulations for both the natural and synthetic models, sweeping the quantity of the input type moi from 1 through 10. Real model: 117 reactions in 61 types. Our synthetic model: 19 reactions in 17 types.

41Discussion Synthesize a design for a precise, robust, programmable probability distribution on outcomes – for arbitrary types and reactions. Computational Synthetic Biology vis-a-vis Technology-Independent Synthesis Implement design by selecting specific types and reactions – say from “toolkit”, e.g. MIT BioBricks repository of standard parts. Experimental Design vis-a-vis Technology Mapping

42Acknowledgements Sponsors: IBM Rochester Blue Gene Development Group NIH “Alpha” Project Center for Genomic Experimentation and Computation (P50 HG02370)

Circuit Modeling Circuit Characterize probability of outcomes. inputsoutputs Model defects, variations, uncertainty, etc.:

Circuit Modeling stochastic logic inputsoutputs Model defects, variations, uncertainty, etc.: 0,1,1,0,1,0,1,1,0,1,… 1,0,0,0,1,0,0,0,0,0,… p 1 = Prob(one) p 2 = Prob(one)

Circuit Modeling stochastic logic inputsoutputs Model defects, variations, uncertainty, etc.: