synthesizing stochasticity in biochemical systems
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Marc Riedel
Synthesizing StochasticitySynthesizing Stochasticityin Biochemical Systemsin Biochemical Systems
Electrical & Computer Engineering
Jehoshua (Shuki) Bruck Caltech
joint work with
Brian Fett Univ. of Minnesota
CIRCUITS & BIOLOGY
RIEDEL lab @ UMN
University of Minnesota
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Engineering novel functionality in biological systems.
BiochemicalReactions
View engineered biochemistry as a form of computation.
Synthetic BiologySynthetic Biology
E. Coli
computationinputs outputs
Molecular Triggers
Molecular Products
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View engineered biochemistry as a form of computation.Bacteria are engineered to produce an anti-cancer drug:
E. Coli
Design ScenarioDesign Scenario
drugtriggering compound
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Bacteria invade the cancerous tissue:
cancerous tissue
Design ScenarioDesign Scenario
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cancerous tissue
The trigger elicits the bacteria to produce the drug:
Design ScenarioDesign Scenario
Bacteria invade the cancerous tissue:
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cancerous tissue
Problem: patient receives too high of a dose of the drug.
Design ScenarioDesign Scenario
The trigger elicits the bacteria produce the drug:
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Design ScenarioDesign Scenario
• Bacteria are all identical.• Population density is fixed.• Exposure to trigger is uniform.
Constraints:
• Control production of drug.Requirement:
Conceptual design problem.
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cancerous tissue
Approach: elicit a fractional response.
Design ScenarioDesign Scenario
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produce drug
triggering compound E. Coli
Approach: engineer a probabilistic response in each bacterium.
with Prob. 0.3
don’t produce drugwith Prob. 0.7
Synthesizing StochasticitySynthesizing Stochasticity
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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 StochasticitySynthesizing Stochasticity
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Generalization: engineer a probability distribution on logical combinations of different outcomes.
cell
A and B with Prob. 0.3
Synthesizing StochasticitySynthesizing Stochasticity
B and C with Prob. 0.7
A with Prob. 0.3
B with Prob. 0.2
C with Prob. 0.5
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Generalization: engineer a probability distribution on logical combinations of different outcomes.
cell
A and B with Prob. 0.3
Synthesizing StochasticitySynthesizing Stochasticity
B and C with Prob. 0.7
Further: program probability distribution with (relative) quantity of input compounds.
)/()Pr( 1 YXfA
)/()Pr( 2 YXfB
)/()Pr( 3 YXfC
X
Y
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CAD Engineers doing BiologyCAD 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?
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Biochemical ReactionsBiochemical Reactions
1 molecule of type A combines with2 molecules of type B to produce2 molecules of type C.( specifies the relative rate of occurrence)k
Reaction
CBA k 22
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Biochemical ReactionsBiochemical Reactions
• Large types (e.g. proteins, enzymes, RNA).• Small quantities (e.g., ~103 molecules/cell).• Complex interactions.
Reaction
VtHBRNAG pZ k CBA k 22
1 molecule of type A combines with2 molecules of type B to produce2 molecules of type C.( specifies the relative rate of occurrence)k
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Discrete AnalysisDiscrete Analysis
“States”
A B C
4 7 5
2 6 822 0 997
S1
S2
S3
A reaction transforms one state into another:
21 1SS Re.g.,
BCAACBCBA
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2 3
2
k1
k2
k3
R1
R2
R3
Track discrete (i.e., integer) quantities of molecular types.
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S1 = [5, 5, 5]
S2 = [4, 7, 4]R1 R2 R3
S3 = [2, 6, 7]
S4 = [1, 8, 6]
Discrete AnalysisDiscrete Analysis
State [A, B, C]
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Discrete AnalysisDiscrete Analysis
BiochemicalReactions
computationinputs outputs
Quantities of Different
Types
Quantities of Different
Types
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BiochemicalReactions
computationinputs outputs
A = 1000B = 333C = 666
A = 0B = 1334C = 226
Quantities of Different
Types
Quantities of Different
Types
Discrete AnalysisDiscrete Analysis
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Probabilistic AnalysisProbabilistic Analysis
The probability that a given reaction is the next to fire is proportional to:
• Its rate constant (i.e., its ki).
• The quantities of its reactants.BCAACBCBA
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2 3
2
k1
k2
k3
R1
R2
R3
See D. Gillespie, “Stochastic Chemical Kinetics”, 2006.
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Probabilistic AnalysisProbabilistic Analysis
Choose the next reaction according to:
jj
iiR
)Pr(
Ri kXnXn 2211
let
2
2
1
1
nX
nX
ki
For each reaction
Probabilistic LatticeProbabilistic 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]
p1p2
p3
p4p5
p6
p7 p8 p9
p10 p11
p12p13
1041 ppp 1393837212625111 )()( ppppppppppppp
[3, 3, 3]start
[3, 3, 3]
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BiochemicalReactions
computationinputs outputs
Quantities of Different
Types
Probability Distribution
on Quantities of Different
Types
Probabilistic ResponseProbabilistic Response
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X = 30Y = 40Z = 30
A with Prob. 0.3
B with Prob. 0.4
C with Prob. 0.3cell
computationinputs outputs
Probabilistic ResponseProbabilistic Response
Quantities of Different
Types
Probability Distribution
on Quantities of Different
Types
BiochemicalReactions
Found in nature? Achievable by design?Yes. Yes.
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Natural StochasticityNatural Stochasticity
Dead Cell
Hijack (Lysis) Stealth (Lysogeny)
“Choice”
Lambda Bacteriophage (Adam Arkin, 1998)
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Prob. 0.2 Prob. 0.8
“Portfolio” of Responses
Natural StochasticityNatural Stochasticity
Dead Cell
“Choice”
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Synthesizing StochasticitySynthesizing 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.
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Synthesizing StochasticitySynthesizing Stochasticity
For types d1, d2, and d3, program the response:Example
SolutionSetup initializing reactions:
Initialize e1, e2, and e3, in the ratio:
30 : 40 : 30
3.01 p 4.02 p 3.03 p
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1 de
21
2 de
31
3 de
29
Setup reinforcing reactions:
Synthesizing StochasticitySynthesizing Stochasticity
For types d1, d2, and d3, program the response:Example
Solution (cont.)
110
11 10023
dde
210
22 10023
dde
310
33 10023
dde
3.01 p 4.02 p 3.03 p
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Setup stabilizing reactions:
For types d1, d2, and d3, program the response:Example
Solution (cont.)
110
21
3
ded
110
31
3
ded 2
1012
3
ded
210
32
3
ded 3
1013
3
ded
310
23
3
ded
Synthesizing StochasticitySynthesizing Stochasticity
3.01 p 4.02 p 3.03 p
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Synthesizing StochasticitySynthesizing Stochasticity
Setup purifying reactions:
Example
Solution (cont.)
For types d1, d2, and d3, program the response:
1021
6
dd 10
31
6
dd 10
32
6
dd
3.01 p 4.02 p 3.03 p
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Result
Synthesizing StochasticitySynthesizing Stochasticity
d1 with Prob. d2 with Prob. d3 with Prob.
Mutually exclusive production of d1, d2, and d3:
Initialize e1, e2, and e3 in the ratio:
x : y : z
zyxzzyx
x zyx
y
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Initializing Reactions
Reinforcing Reactions
Stabilizing
Purifying
Working Reactions
whereijijiii kkkkk ''''
General MethodGeneral Method
ik
i dei i :
ik
ii dedi i 2:'
ik
ji dedij i''
:
'''
: ikji ddij
iik
ii odfdi i ''''
:
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Initializing Reactions
Reinforcing Reactions
Stabilizing
Purifying
Working Reactions
whereijijiii kkkkk ''''
General MethodGeneral Method
ik
i dei i :
ik
ii dedi i 2:'
ik
ji dedij i''
:
'''
: ikji ddij
iik
ii odfdi i ''''
:
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Initializing Reactions
General MethodGeneral Method
ik
i dei i :
For all i, to obtain di with probability pi, select E1, E2,…, En according to:
j jj
iii kE
kEp
Use as appropriate in working reactions:
iik
ii odfdi i ''''
:
(where Ei is quantity of ei)
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Error AnalysisError Analysis
Let
for three reactions (i.e., i, j = 1,2,3).
Require
Performed 100,000 trials of Monte Carlo.
''''''''''ijijiii kkkkk
2'''''''''' ,,1 ijijiii kkkkk
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X03.0
YX 03.002.0
Y03.0
Generalization: engineer a probability distribution with a functional dependence on input quantities.
Functional DependenciesFunctional Dependencies
cell
X
Y
pA 3.0
pB 4.0
pC 3.0
Approach: deterministic “pre-processing”.
AXC 22310 BYA 33
310
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Modular SynthesisModular Synthesis
Deterministic Deterministic ModuleModule
..
..
..Stochastic Stochastic
ModuleModule......
..
..
..
initializing, reinforcing,stabilizing,purifying, and working reactions
linear, exponentiation, logarithm,raising-to-a-power, etc.
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Synthesizing StochasticitySynthesizing 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
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Modeling Natural SystemsModeling 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.
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DiscussionDiscussion
• 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
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AcknowledgementsAcknowledgements
Sponsors:
IBM RochesterBlue Gene Development Group
NIH “Alpha” ProjectCenter for Genomic Experimentation and Computation (P50 HG02370)
Circuit Modeling Circuit Modeling
Circuit
0
1
0
0
1
Characterize probability of outcomes.
inputs outputs
Model defects, variations, uncertainty, etc.:
Circuit Modeling Circuit Modeling
stochastic logic
0
1
0
inputs outputs
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,…
p1 = Prob(one)
p2 = Prob(one)
Circuit Modeling Circuit Modeling
stochastic logic
0
1
0
inputs outputs
Model defects, variations, uncertainty, etc.:
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