REACTION ENUMERATION & CONDENSATION OF DOMAIN-LEVEL STRAND DISPLACEMENT SYSTEMS Stefan Badelt DNA and Natural Algorithms (DNA) Group, Caltech Feb , 2018 14 th 33rd TBI Winterseminar, Bled, Slovenia Grun, Badelt, Sarma, Shin, Wolfe, and Winfree (manuscript in preparation) http://www.github.com/DNA-and-Natural-Algorithms-Group/peppercornenumerator 1
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 verifyably correct artificial systems arbitrary algorithm Anti- Ligand Terminator scalable, correct ON promoter region protein coding region riboswitch components + Ligand - Ligand Terminator OFF information promoter region protein coding region riboswitch conditional switch processing network biological relevance is primary formal description is primary, if experiments fail, refine the method biological relevance secondary 2
DNA STRAND DISPLACEMENT = Adenine = long domain = Thymine = short domain = Cytosine = Guanine = 5' end = Phosphate = 3' end backbone DNA DNA b b a* a* b* b* 3
DOMAIN-LEVEL STRAND DISPLACEMENT long (branch-migration) domain: binds irreversibly short (toehold) domain: binds reversibly A B x x F2 F1 t b a + + t a x t x t b t* x* t* t* x* t* 3-way branch migration unbind bind x x x a x a t b t t t b t* x* t* t* x* t* a t x x t b i3 i1 t* x* t* i2 4
DOMAIN-LEVEL STRAND DISPLACEMENT long (branch-migration) domain: binds irreversibly detailed network short (toehold) domain: binds reversibly condensed network A B x x F2 F1 t b a + + t a x t x t b t* x* t* t* x* t* 3-way branch migration bind unbind x x x a x a t b t t t b t* x* t* t* x* t* a t x x t b i2 i1 t* x* t* 5
DOMAIN-LEVEL STRAND DISPLACEMENT long (branch-migration) domain: binds irreversibly detailed network short (toehold) domain: binds reversibly condensed network A B x x F2 F1 t b a + + t a x t x t b t* x* t* t* x* t* 3-way branch migration unbind bind x x x a x a t b t t t b t* x* t* t* x* t* a t x x t b i2 i1 t* x* t* 6
DOMAIN-LEVEL STRAND DISPLACEMENT long (branch-migration) domain: binds irreversibly short (toehold) domain: binds reversibly A B x x F2 F1 t b a + + t a x t x t b t* x* t* t* x* t* 3-way branch migration unbind bind DSD sytem specification formal CRN x x x a x a t b t t t b formal species: {A, B} signal species (low concentation): {A, B} t* x* t* t* x* t* a t x x t b fuel species (high concentration): {F1, F2} i3 i1 t* x* t* i2 7
FROM CRN TO DSD SYSTEMS Soloveichik Lakin Cardelli (2011) Qian et al. (2011) et al. (2010) et al. (2012) Chen et al. (2012), Cardelli (2013), Srinivas (2015), Lakin et al. (2016), ... Images drawn using VisualDSD, Lakin et al. (2012) 8
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) 9
REACTION TYPES bind / open b a b a* a* a a a a* a* 3-way branch migration 4-way branch migration c c* b b b a a b c c b b* c c* a b b* a* a a* a* b* a* b* c* c* a* a a* b* b a c d* d c b* b c c d* d c* c* a* a* 10
REACTION TYPES allows all secondary structures (pseudoknots excluded) bind / open b open reactions of domains with length > L are forbidden a b a* a* a open & branch migration reactions are always unimolecular, but may lead to dissociation. a a a* bind reactions are the only valid bimolecular reactions a* 3-way branch migration 4-way branch migration c c* b b b a a b c c b b* c c* a b b* a* a a* a* b* a* b* c* c* a* a a* b* b a c d* d c b* b c c d* d c* c* a* a* 11
t t a A B t* b t* 12
t t a A B t* b t* t t a t* b t* 13
t t a t A a B t* b t* t t a t t* b t* 14
t* b t t a t A t* a B t* b t* t t a t t* b t* 15
t* b t t a t A ? t* a B t* b t* t t a t t* b t* 16
t* b t t a t A ? t* a B t* b t* t t a t t* b t* t a t + t* b t* + t a t + t* b t* . . ( + ( . . + ) . ( + ) . ) 17
multistranded pseudoknot t* b t t a t A x ? t* a B t* b t* t t a t t* b t* t a t + t* b t* + t a t + t* b t* . . ( + ( . . + ) . ( + ) . ) 18
SEPARATION OF TIMESCALES unimolecular reactions are fast bimolecular reactions are slow t t a k α t t a A + X B t* b t* k β t* b t* resting complexes transient complex k β k α { X − → A + B ; A + B − → X } at low concentrations: k β [ A ][ B ] << k α [ X ] 19
MODEL PARAMETERS rate-independent model open reactions where domain-length are negligible > L unimolecular reactions are fast bimolecular reactions are slow rate-dependent model assume typical rate constant for every reaction: = rate(rtype, dlength) k unimolecular reactions with are negligible k < k slow unimolecular reactions with are slow k < k fast unimolecular reactions with are fast k ≥ k fast bimolecular reactions are slow 20
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 valid according to enumeration semantics: all valid, except open > L max-helix semantics: reaction types are greedy probability threshold for reactants of bimolecular reactions. probability threshold for products of unimolecular reactions. 21
CRN CONDENSATION Goal: represent CRN in terms of overall slow reactions properties / requirements: 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) 22
CRN CONDENSATION Step 1: Make a graph that contains only fast (1,1) reactions 23
CRN CONDENSATION Step 2: Identify strongly connected components (SCCs) 24
CRN CONDENSATION Step 3: Define transient and resting macrostates A B C D E F 25
CRN CONDENSATION Step 4: Assign fates to complexes (or macrostates) 26
CRN CONDENSATION Step 5: Insert slow reactions & derive condensed reactions 27
DSD CONDENSATION fast (1,1) reaction b a fast (1,2) reaction x x t t b a x t t x t* x* t* t* x* t* slow (2,1) reaction i3 i4 resting macrostate {(B+F1)} {(A+F2)} transient macrostate {} set of fates detailed reactions: F1 F2 A B x x x a x A + F1 -> i1 t b t t b a t i1 -> i2 t* x* t* t* x* t* i2 -> B + F2 B + F2 -> i2 {(A)} {(F1)} {(B)} {(F2)} i2 -> i1 i1 -> A + F1 A + F2 -> i4 i4 -> A + F2 x B + F1 -> i3 x a x a x i3 -> B + F1 t t b t t b condensed reactions: t* x* t* t* x* t* i1 i2 A + F1 -> B + F2 B + F2 -> A + F1 {(A+F1), (B+F2)} 28
REACTION RATE CONDENSATION Consider a condensed reaction: P + Q → K + L + M It is composed of all detailed slow reactions: p + q → I weighted by the decay probability over all pathways: I → ⋯ → k + l + m where p ∈ P , q ∈ Q , k ∈ K , l ∈ L , m ∈ M and is a multiset of intermediate species I 29
REACTION RATE CONDENSATION Notation: detailed reaction: condensed reaction: given : microstate (complex) define : fast (1,1) reaction slow (1,1) reaction then the condensed rate is: fast (1,2) reaction slow (2,1) reaction resting macrostate transient macrostate {} set of fates 30
REACTION RATE CONDENSATION general form: A ̂ k r ̂ = k r ⋅ ℙ [ T B → B ̂ ] ⋅ ℙ [ a i : ] i ∑ ∏ r =( A , B ) ∈ R A ̂ a i ∈ A where stationary distribution A ̂ ℙ [ a i : ] = i reaction decay probability ℙ [ T B → B ̂ ] = 31
A DNA OSCIALLATOR Srinivas, Parkin, Seelig, Winfree, Soloveichik: Enzyme-free nucleic acid dynamical systems. Science (2017) 32
A DNA OSCIALLATOR Srinivas, Parkin, Seelig, Winfree, Soloveichik: Enzyme-free nucleic acid dynamical systems. Science (2017) 33
DETAILED VS. CONDENSED SIMULATION translation scheme: srinivas2017.ts 34
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 35
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