Synthesizing Stochasticity in Biochemical Systems

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


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cancerous tissue

The trigger elicits the bacteria to produce the drug:


Bacteria invade the cancerous tissue:

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cancerous tissue

Problem: patient receives too high of a dose of the drug.


The trigger elicits the bacteria produce the drug:

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• 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.


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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


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cell

A and B with Prob. 0.3


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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cell

A and B with Prob. 0.3


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]


State [A, B, C]

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Quantities of Different

Types


Types

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A = 1000B = 333C = 666

A = 0B = 1334C = 226


Types


Types


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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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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


Probabilistic ResponseProbabilistic Response


Types

Probability Distribution

on Quantities of Different

Types


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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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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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:



Solution (cont.)

110

11 10023

dde

210

22 10023

dde

310

33 10023

dde

3.01 p 4.02 p 3.03 p

30

Setup stabilizing reactions:


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


3.01 p 4.02 p 3.03 p

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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


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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Reinforcing Reactions

Stabilizing

Purifying

Working Reactions

whereijijiii kkkkk ''''


ik

i dei i :

ik

ii dedi i 2:'

ik

ji dedij i''

:

'''

: ikji ddij

iik

ii odfdi i ''''

:

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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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• (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.:


stochastic logic

0

1

0

inputs outputs


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)


stochastic logic

0

1

0

inputs outputs


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Synthesizing Stochasticity in Biochemical Systems

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