[PPT] - MOLECULAR PROGRAMMING (in terms of the nuskell compiler project) PowerPoint Presentation

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REACTION ENUMERATION & CONDENSATION OF DOMAIN-LEVEL STRAND DISPLACEMENT SYSTEMS Stefan Badelt DNA and Natural Algorithms (DNA) Group, Caltech Feb , 2018 33rd TBI Winterseminar, Bled, Slovenia

14th

Grun, Badelt, Sarma, Shin, Wolfe, and Winfree (manuscript in preparation) http://www.github.com/DNA-and-Natural-Algorithms-Group/peppercornenumerator

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MOLECULAR PROGRAMMING

(in terms of the nuskell compiler project) nucleic acids are architecture to implement algorithms chemical reaction networks are a programming language formal/experimental verification of correct implementation

minimal/optimal components for biological systems biological relevance is primary if experiments fail, refine the method verifyably correct artificial systems formal description is primary, biological relevance secondary scalable, correct components arbitrary algorithm conditional switch information processing network

riboswitch protein coding region promoter region Terminator OFF Ligand riboswitch protein coding region promoter region Anti- Terminator ON + Ligand

Ligand

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DNA STRAND DISPLACEMENT

= Adenine = Thymine = Cytosine = Guanine = Phosphate backbone

DNA

= long domain = short domain

DNA

= 3' end = 5' end

b a* b* b b* a*

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DOMAIN-LEVEL STRAND DISPLACEMENT

a t x x t b x* t* t* x b t x t* t* x* t a x t* t* x* t a x b t x t b x* t* t* a t x x t b x* t* t* a t x bind 3-way branch migration unbind

A B F1 F2

short (toehold) domain: binds reversibly long (branch-migration) domain: binds irreversibly

+ +

i1 i2 i3

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DOMAIN-LEVEL STRAND DISPLACEMENT

a t x x t b x* t* t* x b t x t* t* x* t a x t* t* x* t a x b t x t b x* t* t* a t x x t b x* t* t* a t x bind 3-way branch migration unbind

A B F1 F2

short (toehold) domain: binds reversibly long (branch-migration) domain: binds irreversibly

+ +

i1 i2

detailed network condensed network

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DOMAIN-LEVEL STRAND DISPLACEMENT

a t x x t b x* t* t* x b t x t* t* x* t a x t* t* x* t a x b t x t b x* t* t* a t x x t b x* t* t* a t x bind 3-way branch migration unbind

A B F1 F2

short (toehold) domain: binds reversibly long (branch-migration) domain: binds irreversibly

+ +

i1 i2

detailed network condensed network

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DOMAIN-LEVEL STRAND DISPLACEMENT

a t x x t b x* t* t* x b t x t* t* x* t a

A B F1 F2

short (toehold) domain: binds reversibly long (branch-migration) domain: binds irreversibly

+ +

x t* t* x* t a x b t x t b x* t* t* a t x x t b x* t* t* a t x bind 3-way branch migration unbind

i1 i2 i3

formal CRN formal species: {A, B} DSD sytem specification signal species (low concentation): {A, B} fuel species (high concentration): {F1, F2}

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FROM CRN TO DSD SYSTEMS

Cardelli (2011) Soloveichik et al. (2010) Qian et al. (2011) Lakin et al. (2012)

Chen et al. (2012), Cardelli (2013), Srinivas (2015), Lakin et al. (2016), ...

Images drawn using VisualDSD, Lakin et al. (2012)

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FROM A DIGITAL CIRCUIT TO DSD

Qian et al. (2011)

Input for the nuskell compiler: 32 formal reactions. soloveichik2010.ts: 52 signal species, 92 fuel species, 172 intermediate species, 180 reactions. verifies as correct according to the pathway decomposition and CRN bisimulation equivalence Badelt, Johnson, Dong, Shin, Thachuk and Winfree: A general-purpose CRN-to-DSD compiler with formal verification, optimization, and simulation capabilities. LNCS (2017)

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REACTION TYPES

a b a* a*

a

b a* a a a* bind / open b a a* b* b a a* b* b

c c* c* c

3-way branch migration b

b b* a a* c* d* d a a* b b* c b b* a a* c c* d* d a a* b b*

4-way branch migration c* a* c c c* a* c c

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REACTION TYPES

pen & branch migration reactions are always

unimolecular, but may lead to dissociation. bind reactions are the only valid bimolecular reactions

pen reactions of domains with length > L are forbidden

allows all secondary structures (pseudoknots excluded)

a b a* a*

a

b a* a a a* bind / open b a a* b* b a a* b* b

c c* c* c

3-way branch migration b

b b* a a* c* d* d a a* b b* c b b* a a* c c* d* d a a* b b*

4-way branch migration c* a* c c c* a* c c

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a t t t* t* b A B

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a t t t* t* b A B a t t t* t* b

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a t t t* t* b A B a t t t* t* b a t t

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a t t t* t* b A B a t t t* t* b a t t t* t* b

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a t t t* t* b A B a t t t* t* b a t t t* t* b ?

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a t t t* t* b A B a t t t* t* b a t t t* t* b ? t a t + t* b t* + t a t + t* b t* . . ( + ( . . + ) . ( + ) . )

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a t t t* t* b A B a t t t* t* b a t t t* t* b ? t a t + t* b t* + t a t + t* b t* . . ( + ( . . + ) . ( + ) . )

x

multistranded pseudoknot

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SEPARATION OF TIMESCALES

unimolecular reactions are fast bimolecular reactions are slow

a t t t* t* b A B a t t t* t* b kα kβ + transient complex resting complexes X

{X A + B; A + B X} −→

kα

− →

kβ

at low concentrations:

[A][B] << [X] kβ kα

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MODEL PARAMETERS

rate-independent model

pen reactions where domain-length

are negligible unimolecular reactions are fast bimolecular reactions are slow

> L

rate-dependent model assume typical rate constant for every reaction: = rate(rtype, dlength) unimolecular reactions with are negligible unimolecular reactions with are slow unimolecular reactions with are fast bimolecular reactions are slow

k k < kslow k < kfast k ≥ kfast

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valid according to enumeration semantics:

REACTION ENUMERATION

every complex has all valid fast reactions enumerated transient complexes have no slow reactions enumerated resting complexes have all valid slow reactions enumerated all initial complexes are included all valid, except open max-helix semantics: reaction types are greedy probability threshold for reactants of bimolecular reactions. probability threshold for products of unimolecular reactions.

> L

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CRN CONDENSATION Goal: represent CRN in terms of overall slow reactions

all fast reactions are unimolecular reactions have arity (n,m) with n > 0 and m > 0 reactants of slow reactions must be resting states reactants and products of fast (1-2) reactions are in different SCCs (mass conservation)

properties / requirements:

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CRN CONDENSATION Step 1: Make a graph that contains only fast (1,1) reactions

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CRN CONDENSATION Step 2: Identify strongly connected components (SCCs)

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CRN CONDENSATION Step 3: Define transient and resting macrostates

C D E F A B

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CRN CONDENSATION Step 4: Assign fates to complexes (or macrostates)

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CRN CONDENSATION Step 5: Insert slow reactions & derive condensed reactions

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DSD CONDENSATION

a t x

A

x t b x* t* t*

F1

x t* t* x* t a

F2

x t b x* t* t* a t x

i1

x t* t* x* t a x b t

i2

x t* t* x* t a a x t

i4

x t b x* t* t* x t b

i3

x b t

B

{(A+F2)} {(B+F1)} {(A)} {(F1)} {(B)} {(F2)} {(A+F1), (B+F2)} fast (1,1) reaction fast (1,2) reaction slow (2,1) reaction resting macrostate transient macrostate

{}

set of fates

condensed reactions:

A + F1 -> B + F2 B + F2 -> A + F1

detailed reactions:

A + F1 -> i1 i1 -> i2 i2 -> B + F2 B + F2 -> i2 i2 -> i1 i1 -> A + F1 A + F2 -> i4 i4 -> A + F2 B + F1 -> i3 i3 -> B + F1

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REACTION RATE CONDENSATION

Consider a condensed reaction:

P + Q K + L + M →

It is composed of all detailed slow reactions: weighted by the decay probability over all pathways:

p + q I → I → ⋯ → k + l + m

where and is a multiset of intermediate species

p ∈ P, q ∈ Q, k ∈ K, l ∈ L, m ∈ M I

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REACTION RATE CONDENSATION

microstate (complex) fast (1,1) reaction slow (1,1) reaction fast (1,2) reaction slow (2,1) reaction resting macrostate transient macrostate

{}

set of fates

detailed reaction: condensed reaction:

Notation: given: define: then the condensed rate is:

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REACTION RATE CONDENSATION

general form:

= ⋅ ℙ[ ] ⋅ ℙ[ : ] kr̂ ∑

r=(A,B)∈RÂ

kr TB→B̂ ∏

∈A ai

ai Â

i

where stationary distribution reaction decay probability

ℙ[ : ] = ai Â

i

ℙ[ ] = TB→B̂

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A DNA OSCIALLATOR

Srinivas, Parkin, Seelig, Winfree, Soloveichik: Enzyme-free nucleic acid dynamical systems. Science (2017)

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A DNA OSCIALLATOR

Srinivas, Parkin, Seelig, Winfree, Soloveichik: Enzyme-free nucleic acid dynamical systems. Science (2017)

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DETAILED VS. CONDENSED SIMULATION

translation scheme: srinivas2017.ts

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REACTION ENUMERATOR

model limitations no multistranded pseudoknots assumption of low concentrations assumption of "typical" reaction rate constants model parameters multiple layers of reaction-semantics reaction types max-helix notion (representation-independent) reaction rate dependent enumeration

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What the domain level can do: enumerate intended reaction pathways detect unintended reaction pathways very fast assessment of overall dynamics define a CRN for sequence-level simulations What the domain level cannot do: include sequence-level variations within the domains What the domain level could do: detect and quantify particular leak reactions provide a coarse-graining for stochastic simulations

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THANKS TO

Erik Winfree Casey Grun Karthik Sarma Seung Woo Shin you Brian Wolfe

http://www.github.com/DNA-and-Natural-Algorithms-Group/peppercornenumerator This research was funded in parts by: The Caltech Biology and Biological Engineering Division Fellowship. The U.S. National Science Foundation NSF Grant CCF-1213127 and NSF Grant CCF-1317694. The Gordon and Betty Moore Foundation's Programmable Molecular Technology Initiative (PMTI).