Synthesizing Stochasticity in Biochemical Systems

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Synthesizing Stochasticityin Biochemical Systems
Marc Riedel
Electrical Computer Engineering
University of Minnesota
CIRCUITS BIOLOGY
joint work with
Jehoshua (Shuki) Bruck
Caltech
Brian Fett
Univ. of Minnesota
RIEDEL lab _at_ UMN
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Synthetic Biology
Engineering novel functionality in biological
systems.
View engineered biochemistry as a form of
computation.
computation
inputs
outputs
Biochemical Reactions
Molecular Triggers
Molecular Products
E. Coli
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Design Scenario
View engineered biochemistry as a form of
computation.
Bacteria are engineered to produce an anti-cancer
drug
triggering compound
drug
E. Coli
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Design Scenario
Bacteria invade the cancerous tissue
cancerous tissue
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Design Scenario
The trigger elicits the bacteria to produce the
drug
Bacteria invade the cancerous tissue
cancerous tissue
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Design Scenario
The trigger elicits the bacteria produce the
drug
Problem patient receives too high of a dose of
the drug.
cancerous tissue
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Design Scenario
Conceptual design problem.
Constraints

• Bacteria are all identical.
• Population density is fixed.
• Exposure to trigger is uniform.

Requirement

• Control production of drug.

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Design Scenario
Approach elicit a fractional response.
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Synthesizing Stochasticity
Approach engineer a probabilistic response in
each bacterium.
produce drug
with Prob. 0.3
triggering compound
dont produce drug
with Prob. 0.7
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Synthesizing Stochasticity
Generalization engineer a probability
distribution on logical combinations of different
outcomes.
A
with Prob. 0.3
B
with Prob. 0.2
cell
C
with Prob. 0.5
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Synthesizing Stochasticity
Generalization engineer a probability
distribution on logical combinations of different
outcomes.
A
with Prob. 0.3
B
with Prob. 0.2
cell
C
with Prob. 0.5
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Synthesizing Stochasticity
Generalization engineer a probability
distribution on logical combinations of different
outcomes.
X
Y
cell
Further program probability distribution with
(relative) quantity of input compounds.
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CAD Engineers doing Biology
Why?

• Specific computational expertise

with data structures and algorithms for analyzing
and manipulating discrete designs over a large
state space.
How?

• Cast problems in a computational language

with well-defined, quantitative inputs and
outputs tackling analysis and synthesis
systematically.
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Biochemical Reactions
1 molecule of type A combines with 2 molecules
of type B to produce 2 molecules of type C.
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Biochemical Reactions
Reaction
1 molecule of type A combines with 2 molecules
of type B to produce 2 molecules of type C.

• Large types (e.g. proteins, enzymes, RNA).
• Small quantities (e.g., 103 molecules/cell).
• Complex interactions.

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Discrete Analysis
Track discrete (i.e., integer) quantities of
molecular types.
States
A
B
C
S1
4
7
5
S2
2
6
8
S3
22
0
997
A reaction transforms one state into another
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Discrete Analysis
S1 5, 5, 5
R1
R2
R3
S2 4, 7, 4
State A, B, C
S3 2, 6, 7
S4 1, 8, 6
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Discrete Analysis
computation
inputs
outputs
Quantities of Different Types
Quantities of Different Types
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Discrete Analysis
computation
inputs
outputs
Quantities of Different Types
Quantities of Different Types
A 1000
A 0
B 333
B 1334
C 666
C 226
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Probabilistic 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.

See D. Gillespie, Stochastic Chemical Kinetics,
2006.
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Probabilistic Analysis
For each reaction
let
Choose the next reaction according to
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Probabilistic Lattice
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Probabilistic Response
computation
inputs
outputs
Probability Distribution on Quantities of
Different Types
Quantities of Different Types
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Probabilistic Response
computation
inputs
outputs
Probability Distribution on Quantities of
Different Types
X 30
Quantities of Different Types
Y 40
Z 30
Found in nature?
Achievable by design?
Yes.
Yes.
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Natural Stochasticity
Lambda Bacteriophage (Adam Arkin, 1998)
Hijack (Lysis)
Stealth (Lysogeny)
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Natural Stochasticity
Portfolio of Responses
Prob. 0.2
Prob. 0.8
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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.

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Synthesizing Stochasticity
Example
For types d1, d2, and d3, program the response
Solution
Setup initializing reactions
Initialize e1, e2, and e3, in the ratio
30 40 30
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Synthesizing Stochasticity
Example
For types d1, d2, and d3, program the response
Solution (cont.)
Setup reinforcing reactions
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Synthesizing Stochasticity
Example
For types d1, d2, and d3, program the response
Solution (cont.)
Setup stabilizing reactions
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Synthesizing Stochasticity
Example
For types d1, d2, and d3, program the response
Solution (cont.)
Setup purifying reactions
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Synthesizing Stochasticity
Initialize e1, e2, and e3 in the ratio
x y z
Result
Mutually exclusive production of d1, d2, and d3
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General Method
Initializing Reactions
Reinforcing Reactions
Stabilizing
Purifying
Working Reactions
where
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General Method
Initializing Reactions
Reinforcing Reactions
Stabilizing
Purifying
Working Reactions
where
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General Method
Initializing Reactions
For all i, to obtain di with probability pi,
select E1, E2,, En according to
(where Ei is quantity of ei)
Use as appropriate in working reactions
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Error Analysis
Require
Let
for three reactions (i.e., i, j 1,2,3).
Performed 100,000 trials of Monte Carlo.
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Functional Dependencies
Generalization engineer a probability
distribution with a functional dependence on
input quantities.
X
Y
cell
Approach deterministic pre-processing.
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Modular Synthesis
initializing, reinforcing,stabilizing, purifying
, and working reactions
linear, exponentiation, logarithm,raising-to-a-p
ower, etc.
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Synthesizing Stochasticity
Synthesizing Stochasticity in Biochemical Systems

• (potential) Applicationsbiochemical sensing,
  drug production, disease treatment.
• (immediate) Impetus framework for analyzing and
  characterizing the stochastic behavior of natural
  biological systems.

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Modeling Natural Systems
Lambda Bacteriophage (Adam Arkin, 1998)

• Real model 117 reactions in 61 types.
• Our synthetic model 19 reactions in 17 types.

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.
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Discussion

• Synthesize a design for a precise, robust,
  programmable probability distribution on outcomes
  for arbitrary types and reactions.

• Implement design by selecting specific types and
  reactions say from toolkit, e.g. MIT
  BioBricks repository of standard parts.

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Acknowledgements
Sponsors
IBM RochesterBlue Gene Development Group
NIH Alpha ProjectCenter for Genomic
Experimentation and Computation (P50 HG02370)
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Circuit Modeling
Model defects, variations, uncertainty, etc.
0
0
1
1
0
Characterize probability of outcomes.
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Circuit Modeling
Model defects, variations, uncertainty, etc.
p1 Prob(one)
0
0,1,1,0,1,0,1,1,0,1,
1
1,0,0,0,1,0,0,0,0,0,
0
p2 Prob(one)
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Circuit Modeling
Model defects, variations, uncertainty, etc.
0
1
0