equivalence of chemical reaction networks in a crn …badelt/files/slides/vemdp... · 2018. 7....

1

EQUIVALENCE OF CHEMICAL REACTION NETWORKS

IN A CRN-TO-DNA COMPILER FRAMEWORK and Erik WinfreeStefan Badelt

DNA and Natural Algorithms (DNA) Group, Caltech

Oxford, July, 19 , 2018th

VEMDP 2018

http://dna.caltech.edu/~badelt

2

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 switchinformationprocessingnetwork

riboswitchprotein coding regionpromoter region

Term

inat

or

OFF

Ligand

riboswitchprotein coding regionpromoter region

Anti-Terminator

ON

+ Ligand

- Lig

and

3

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*

4

DOMAIN-LEVEL STRAND DISPLACEMENT

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

x

t*t* x*

tax

btx t b

x* t*t*

a tx

x t b

x* t*t*

a t x

bind3-way branch migrationunbind

A BF1 F2

short (toehold) domain: binds reversiblylong (branch-migration) domain: binds irreversibly

+ +

i1

i2

i3

5


a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

x

t*t* x*

tax

btx t b

x* t*t*

a tx

x t b

x* t*t*

a t x

bind 3-way branch migration unbind

A BF1 F2


+ +

i1 i2

detailed networkcondensed network

6


a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

x

t*t* x*

tax

btx t b

x* t*t*

a tx

x t b

x* t*t*

a t x


A BF1 F2


+ +

i1 i2

detailed networkcondensed network

7


a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2


+ +

x

t*t* x*

tax

btx t b

x* t*t*

a tx

x t b

x* t*t*

a t x


i1

i2

i3

formal CRN

formal species: {A, B}

DSD sytem specification

signal species (low concentation): {A, B}fuel species (high concentration): {F1, F2}

8

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)

9

A CRN-TO-DSD COMPILERtry all translation schemes

verify correctness

sequence design experimental

testing

chooseoptimal scheme

automatedworkflow

- for all inputs- experimental setup

- synthesis cost- efficency / robustness- experimental constraints

10

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, Shin, Johnson, Dong, Thachuk and Winfree: A general-purpose CRN-to-DSD compiler withformal verification, optimization, and simulation capabilities. LNCS (2017)

11

THE COMPILER FRAMEWORK

a txA

xbtB

x t b

x* t*t* F1

x

t*t* x*

taF2

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +

CRN trajectory equivalence

CRN condensation

CRN enumeration

1

2

3

nucleotidesequences

4

x

t*t* x*

tax

btx t b

x* t*t*

a tx

x t b

x* t*t*

a t x

bind

3-way branch migration

unbind

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +

i1

i2

i3

KinDA project

nucleotide-level reaction rates

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +

Nuskell project Peppercorn project

a txA

+x

t*t* x*

taF2

x

t*t* x*

taa

xt

condensed reaction rates

domain-level reaction rates

Badelt et al. (2017) - Nuskell Grun et al. (2014) - Peppercorn

Shin et al. (2017) - CRN pathway decomposition equivalence Johnson et al. (2018) - CRN bisimulation equivalence

Berleant et al. (submitted) - KinDA

12

REACTION ENUMERATION

a b a* a*a

b

a*

a a

a*

bind / open

ba

a* b*

ba

a* b*

bc

c*c*

c


bb

b* a

a*c*

d* d

a

a* b

b*c b

b*

a

a*

c c*

d* d

a

a*

b

b*


c*a*

cc

c*a*

cc

13

REACTION ENUMERATION

open & branch migration reactions are always unimolecular, but may lead to dissociation.

bind reactions are the only valid bimolecular reactions

open reactions of domains with length > L are forbiddenallows all secondary structures (pseudoknots excluded)

a b a* a*a

b

a*

a a

a*

bind / open

ba

a* b*

ba

a* b*

bc

c*c*

c


bb

b* a

a*c*

d* d

a

a* b

b*c b

b*

a

a*

c c*

d* d

a

a*

b

b*


c*a*

cc

c*a*

cc

14

SEPARATION OF TIMESCALESunimolecular reactions are fast bimolecular reactions are slow

a tt

t* t*b

A

B

a tt

t* t*b

kα

kβ

+

transient complexresting complexes

X

{X A + B; A + B X}−→kα

−→

kβ

at low concentrations:

[A][B]

15

APPROXIMATE REACTION RATE CONSTANTS

a*

a a

a*bimolecular binding

open rate

h

a a*a a*h

ha

a*linker closing(nucleation)

hairpin closing rate

helix zipping

unimolecular binding

branch migration

ba

a* b*

ba

a* b*

bc

c*c*

cb

c*a*

cc

c*a*

cc

16

MODEL PARAMETERSnegligible reactions slow reactions fast reactions

bimolecular [/M/s]

unimolecular [/s]

bind21

open (len > L) open (len < L)bind11branch migration

unimolecular [/s]

rate-independent model: simple, one parameter: Lrate-dependent model: flexible, two parameters: k-slow, k-fast

17

valid according to enumeration semantics:

REACTION ENUMERATIONall initial complexes are includedevery complex has all valid fast reactions enumeratedtransient complexes have no slow reactions enumeratedresting complexes have all valid slow reactions enumerated

rate-dependent modelrate-independent modelmax-helix semantics: reaction types are greedyreject-remote semantics: exclude remote-toehold branch migration

18


a txA

xbtB

x t b

x* t*t* F1

x

t*t* x*

taF2

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +


CRN condensation

CRN enumeration

1

2

3

nucleotidesequences

4

x

t*t* x*

tax

btx t b

x* t*t*

a tx

x t b

x* t*t*

a t x

bind


unbind

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +

i1

i2

i3

KinDA project


a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +


a txA

+x

t*t* x*

taF2

x

t*t* x*

taa

xt






19

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:

20

CRN CONDENSATION

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

21

CRN CONDENSATION

Step 2: Identify strongly connected components (SCCs)

22

CRN CONDENSATION

Step 3: Define transient and resting macrostates

C D E F

A B

23

CRN CONDENSATION

Step 4: Assign fates to complexes (or macrostates)

24

CRN CONDENSATION

Step 5: Insert slow reactions & derive condensed reactions

25

DSD CONDENSATION

a txA x t b

x* t*t*

F1x

t*t* x*

taF2

x t b

x* t*t*

a tx

i1

x

t*t* x*

tax

bt

i2

x

t*t* x*

taa

xt

i4

x t b

x* t*t*

x tb

i3

xbt

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 + F2B + F2 -> A + F1

detailed reactions:A + F1 -> i1i1 -> i2i2 -> B + F2B + F2 -> i2i2 -> i1i1 -> A + F1A + F2 -> i4i4 -> A + F2B + F1 -> i3i3 -> B + F1

26

REACTION RATE CONDENSATIONConsider 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

27

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:

28

REACTION RATE CONDENSATIONgeneral form:

= ⋅ ℙ[ ] ⋅ ℙ[ : ]kr ̂ ∑r=(A,B)∈R

Â

kr TB→B̂ ∏∈Aai

ai Â i

where

stationary distribution reaction decay probability

ℙ[ : ] =ai Â iℙ[ ] =TB→B̂

29

EXPERIMENTAL DATA: YIN ET AL. (2008)

C.D

C

D

A.B

A.B.C

(3)

(4)C.D.A

(5)

(6)

I

(1) (2a)

Exponentialamplification

Initiation

I.A

A

Autocatalys is

B

A

0.001×

0× (No initiator)0.0005× (10 pM initiator)

0.003×

0.01×0.005×1× 0.3× 0.1× 0.05× 0.03× 0.02×

Aa a*b

c*

x y

b*x* y*

x*v

v* C

c dy v

d*y* v*

c*

a*

u*

u*

u

x*

D

dd*

a cx u

a* c*x* u*

v*

b*

v

v*y*

Ia* x* b* y*v*Metas tablemonomers Initiator

bb*

c ay x

c*a*y* x*

y*

d*u*

u

*vB

u*

ba

10% completion

v* y*

B

A B

D C

(1) (2b)

(3)

(4a)

(4b)(5)

(6b)

(2a) (6a)

I

2

3

4

0 1 2 3 4 5

Tim

e (h

)

1.5

c

Time (h)

Emis

sion

s (c

ount

s s–

1 )

×105

log10 [I]–1.6–2–2.4 –1.2

1.2

0.8

0.4A+B+C+D

A+B A

I.A.B(2b)

30

EXPERIMENTAL DATA: KOTANI & HUGHES (2017)

31

MORE EXPERIMENTAL DATA (2009 - 2017)

32


a txA

xbtB

x t b

x* t*t* F1

x

t*t* x*

taF2

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +


CRN condensation

CRN enumeration

1

2

3

nucleotidesequences

4

x

t*t* x*

tax

btx t b

x* t*t*

a tx

x t b

x* t*t*

a t x

bind


unbind

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +

i1

i2

i3

KinDA project


a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +


a txA

+x

t*t* x*

taF2

x

t*t* x*

taa

xt






33

CRN EQUIVALENCE

A -> A + AA + A -> AA + B -> B + BB ->A + C ->C -> C + CC + C -> C

translation scheme: qian2011_3D_var1.ts

34

CRN EQUIVALENCE


f14 + C -> e1428 + f15e853 + f12 -> C + f13A + f4 -> f3 + e71f2 + e25 -> A + f1A + e25 -> f2 + e7e996 + f3 -> A + f10e1428 + f15 -> f14 + Cf3 + e71 -> A + f4e465 + B -> e418 + f6e614 + f9 -> e611 + e730e996 + C -> e1040 + f12e465 + f5 -> e514 + e368e308 + f7 -> f8 + Be418 + B -> e371 + f6C + f13 -> e853 + f12A + f1 -> f2 + e25B + e71 -> e319 + f7f2 + e7 -> A + e25e1040 + f11 -> e1162 + e1163 + e1158e319 + f7 -> B + e71e308 + f9 -> e614 + e615 + e611e371 + f6 -> e418 + Be1040 + f12 -> e996 + Cf8 + B -> e308 + f7e319 + f5 -> e372 + e371 + e368f3 + e7 -> A + f0e853 + f15 -> e1428 + Ce1428 + C -> e853 + f15A + f10 -> e996 + f3e1163 + f11 -> e1158 + e1246e418 + f6 -> e465 + BA + f0 -> f3 + e7


35

CRN EQUIVALENCE



C -> e1428e853 -> CA -> e71e25 -> AA + e25 -> e7e996 -> Ae1428 -> Ce71 -> Ae465 + B -> e418e614 -> e611 + e730e996 + C -> e1040e465 -> e514 + e368e308 -> Be418 + B -> e371C -> e853A -> e25B + e71 -> e319e7 -> A + e25e1040 -> e1162 + e1163 + e1158e319 -> B + e71e308 -> e614 + e615 + e611e371 -> e418 + Be1040 -> e996 + CB -> e308e319 -> e372 + e371 + e368e7 -> Ae853 -> e1428 + Ce1428 + C -> e853A -> e996e1163 -> e1158 + e1246e418 -> e465 + BA -> e7


36

CRN EQUIVALENCE


C -> e1428e853 -> CA -> e71e25 -> AA + e25 -> e7e996 -> Ae1428 -> Ce71 -> Ae465 + B -> e418e614 -> e611 + e730e996 + C -> e1040e465 -> e514 + e368e308 -> Be418 + B -> e371C -> e853A -> e25B + e71 -> e319e7 -> A + e25e1040 -> e1162 + e1163 + e1158e319 -> B + e71e308 -> e614 + e615 + e611e371 -> e418 + Be1040 -> e996 + CB -> e308e319 -> e372 + e371 + e368e7 -> Ae853 -> e1428 + Ce1428 + C -> e853A -> e996e1163 -> e1158 + e1246e418 -> e465 + BA -> e7

C -> CC + C -> CA -> AA -> AA + A -> A + AA -> AC -> CA -> AB -> B -> A + C -> A + C ->B -> BB + B -> B + BC -> C + CA -> AB + A -> A + BA + A -> A + AA + C ->A + B -> B + AB -> B + B -> B + BA + C -> A + CB -> BA + B -> B + BA + A -> AC + C -> C + CC + C -> C + CA -> A ->B -> BA -> A + A

A => A B => B C => C e1040 => A, C e1158 => e1162 => e1163 => e1246 => e1428 => C e25 => A e308 => B e319 => A, B e368 => e371 => B, B e372 => e418 => B e465 => e514 => e611 => e614 => e615 => e7 => A, A e71 => A e730 => e853 => C, C e996 => AIn

terp

reta

tion

(CR

N-b

isim

ulat

ion)

:


Johnson et al. (2016) - CRN bisimulation equivalence translation scheme: qian2011_3D_var1.ts

37


a txA

xbtB

x t b

x* t*t* F1

x

t*t* x*

taF2

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +


CRN condensation

CRN enumeration

1

2

3

nucleotidesequences

4

x

t*t* x*

tax

btx t b

x* t*t*

a tx

x t b

x* t*t*

a t x

bind


unbind

a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +

i1

i2

i3

KinDA project


a tx

x t b

x* t*t*

xbtx

t*t* x*

ta

A BF1 F2+ +


a txA

+x

t*t* x*

taF2

x

t*t* x*

taa

xt






38

KINETIC NUCLEOTIDE-LEVEL ANALYSIS

A

B

121* 1*2*1

2

domain level sequence level

544

35*

212

54

1 2

435*

2

544

35*

2

435*

212

domain level

sequence level

544

35*

212

54

1 2

435*

2535*

1

Berleant, Berlind, Badelt, Dannenberg, Schaeffer, Winfree (submitted) - Automated Sequence-LevelAnalysis of Kinetics and Thermodynamics for Domain-Level DNA Strand-Displacement Systems

39

KINETIC NUCLEOTIDE-LEVEL ANALYSIS

S1

C1

P2

P3

Rd1s* d2* t3*

d1s d2

t2

ab

b*

a*

c*c

t1*

T2

d1s

t2*

act1

t1*b c*

cb*

d1s

T2 b a t2

t2*ab*d2

t3

time, hr

conc

entra

tion,

nM

conc

entra

tion,

nM -

time, s

P1act1

a*c*t1* t2*

I1

t2

ab

b*c

T2

d1s

S2

t1*

a*b*

b

a

c c*

t2*

t1

d2

t3

Dd1s d2

RW

t3

b

b*

d1s*

t3*

d2d2* a*

d1s

T2 a t2

t2*

k1 = (5.23 ± 1.69) x 105 M-1s-1

k2 = (174 ± 28) s-1

k1 < 78.7 M-1s-1

k1 = (9.49 ± 2.92) x 105 M-1s-1

k2 = (0.66 ± 0.13) s-1

k1 = (1.33 ± 0.22) x 106 M-1s-1

k2 = (2.65 ± 0.31) s-1

S2-R

d2

t3

d1s*d2*t3*

d1sd2

t1*

a*b*

b

a

c c*

t2*

t1

d2

t3

d1s*d2*

t3*

d1sd2

t1*

a*b*

b

a

c c*

t2*

t1

e51

e48

S2 R

C1 P3 RWD

I1S2-R

k1 = (3.18 ± 0.30) x 107 M-1s-1

k2 = (1.57 ± 0.26) x 107 s-1

k1 = (1.14 ± 0.43) x 105 M-1s-1

k2 = (1.14 ± 0.24) s-1

A C

DB DOMAIN SEQUENCECAGTCCCAAGTCACCACCTAGCGCACTCGCGATACGAGGCCTGG

CCGTTT

CCAGATCAGCAGCCATTCGTTC

ACATCC

CCTCTACTCA

ba

ct1t2t3

CCAAACCTTCATCTTC

TACTCG

d1sTTT2

d2

Berleant, Berlind, Badelt, Dannenberg, Schaeffer, Winfree (submitted) - Automated Sequence-LevelAnalysis of Kinetics and Thermodynamics for Domain-Level DNA Strand-Displacement Systems

40

THANKS TO

Erik Winfree Seung WooShin

Casey Grun JosephBerleant

Robert F.Jonhson

ChrisThachuk

FritsDannenberg

Chris Berlind KarthikSarma

Qing Dong JosephSchaeffer

Brian Wolfe

http://www.github.com/DNA-and-Natural-Algorithms-Group/nuskellhttp://www.github.com/DNA-and-Natural-Algorithms-Group/peppercornenumerator

http://www.github.com/DNA-and-Natural-Algorithms-Group/KinDAThis 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).
http://www.github.com/DNA-and-Natural-Algorithms-Group/nuskellhttp://www.github.com/DNA-and-Natural-Algorithms-Group/peppercornenumeratorhttp://www.github.com/DNA-and-Natural-Algorithms-Group/kinda

equivalence of chemical reaction networks in a crn …badelt/files/slides/vemdp... · 2018. 7....

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