Article for analog vector Article for analog vector algebra computationalgebra computation

Allen P. Mils Jr, Bernard Yurke, PhiAllen P. Mils Jr, Bernard Yurke, Philip M Platzmanlip M Platzman

IntroductionIntroduction

► The chemical operations that can be performed on The chemical operations that can be performed on strands of DNA can be exploited to represent variostrands of DNA can be exploited to represent various ordinary algebraic operations including mathemus ordinary algebraic operations including mathematical algorithm.atical algorithm.

► Chemical OperationsChemical Operations LigationLigation PolymerasePolymerase Cutting via restriction enzymesCutting via restriction enzymes Base-specific hybridizationBase-specific hybridization Melting of duplex DNAMelting of duplex DNA Destruction of DNADestruction of DNA

IntroductionIntroduction

►ObjectivesObjectives An analog representation for the operations of An analog representation for the operations of

vector algebra including inner and outer products vector algebra including inner and outer products of dyads and vectors.of dyads and vectors.

► StrategiesStrategies Applying Oliver’s work(1997)Applying Oliver’s work(1997)

► Including a representation for negative real number as Including a representation for negative real number as well as positive real number represented in Oliver’s well as positive real number represented in Oliver’s work.work.

Representing the Hopfield associative memory Representing the Hopfield associative memory and the feed-forward neural network. and the feed-forward neural network.

Oliver’s WorkOliver’s Work► Design of DNA sequences for multiplication of

two matrices.► A Multiplication of matrices X and Y.► B The graph representation of the operation. The

row and column identifiers of the matrices are represented by vertices (circles) in the graph. Initial and terminal vertices are red and intermediate vertices are green. Nonzero elements in matrices X, Y, and Z are represented by directed edges (arrows) which connect vertices for the appropriate row and column identifying that element. Thus the directed edge connecting vertices ‘‘1’’ and ‘‘a’’ in the graph represents the symbol ‘‘1’’ in row 1 column a of X. The graph representation of Z also can be drawn by inspection of the matrix. Alternatively, the graph of Z can be determined from the graph representing the product of X and Y. The edges in Z represent paths [in this case, a path is simply a sequence of edges that connect an initial vertex (1, 2, or 3) to a terminal vertex (A or B)] between initial and terminal vertices in the graph on the left. Z is constructed by replacing the paths in (X)(Y) with edges and removing all intermediate vertices (a, b, c, and d). Thus the edge from 1 to B in the graph of Z represents the path 1-a, a-B in the graph for (X)(Y).

► C The DNA strands used to represent the nonzero elements (edges in the graph) of each matrix. The ends of the DNA sequences represent vertices at either end of the respective edge in the graph. Thus 1, 2, and 3 label restriction enzyme sites which represent the initial vertices 1, 2, and 3. A and B label restriction enzyme sites which represent the terminal vertices A and B. The intermediate vertices for which an edge is entering the vertex are represented by the single-stranded DNA sequences a, b, c, or d. Intermediate vertices for which an edge is exiting the vertex are represented by the complementary sequences a8, b8, c8, and d8, respectively.

Oliver’s WorkOliver’s Work► Reaction sequence used to multiply matrices X

and Y.► The desired DNA strands are synthesized such t

hat the sequences representing initial and terminal vertices are double stranded. The intermediate vertices are represented by single-strand overhangs. The DNA sequences are mixed, annealed, and ligated in a reaction that generates all possible (in this case, four) paths. The reaction mixture is divided into six equal aliquots which are used in separate restriction enzyme digest reactions. Each aliquot represents an element in the product matrix. To each aliquot is added two restriction enzymes. One enzyme corresponds to the row of the product matrix that the element occupies, and the other restriction enzyme corresponds to the column that the element occupies. The paths in each of these six reactions will be either uncut, cut once, or cut at both ends depending on the restriction sites incorporated in the path. A portion of each of the restriction enzyme reactions is submitted to gel electrophoresis which separates the strands based on size. Paths which have been cut by each of the enzymes in a particular reaction, thus representing the symbol ‘‘1’’ for that element, will appear as bands on the gel.

Oliver’s WorkOliver’s Work

► The square and cube of a matrix and the graph representations of the operations.

Oliver’s WorkOliver’s Work

► The graph representation for the multiplication of two matrices containing real, positive numbers. The numbers over the edges are the transmission factors for each edge.

'3'5 AGCTATCGATEi

DNA Vector spaceDNA Vector space

► VectorsVectors Basis vector and vector spaceBasis vector and vector space

Concentration [EConcentration [Eii]]►Represented by a DNA sample containing ERepresented by a DNA sample containing E ii strands with concent strands with concent

ration [Eration [Eii] proportional to the amplitude V] proportional to the amplitude V ii..

Basis vector eBasis vector eii

10-dimensional Vector space10-dimensional Vector space

m

iiieVV

1

Amplitude of the i-th component of the vectorAmplitude of the i-th component of the vector

DNA Vector spaceDNA Vector space

► VectorsVectors Practical choicePractical choice

'3.........'5 2132121 GTASSSNNNNRRTACRE riq

iiiri

Palindromic restriction enzyme(Bst1107I)Palindromic restriction enzyme(Bst1107I)recognition sequencerecognition sequence

Invariant r-mersInvariant r-mersTo assist in hybridization operationTo assist in hybridization operation

A, G, C, TA, G, C, T

DNA Vector spaceDNA Vector space

► VectorsVectors Negative vectorsNegative vectors

►Since concentrations are always positive, we need an appropriatSince concentrations are always positive, we need an appropriate representation for negative amplitudes.e representation for negative amplitudes.

►We choose to represent negative unit vectors eWe choose to represent negative unit vectors e ii by the sequence by the sequence of bases complementary to Eof bases complementary to Eii..

►As a result, when two vectors are added, any positive and negatiAs a result, when two vectors are added, any positive and negative amplitude will hybridize and can be removed by digestion usinve amplitude will hybridize and can be removed by digestion using a suitable enzyme or by column separation.g a suitable enzyme or by column separation.

'3...............'5 1212312 GTARRRNNNNSSSTACE riiii

qri

Addition of VectorsAddition of Vectors

► Combine in one container equal quantities from the two collCombine in one container equal quantities from the two collections of DNA representing the two vectors at twice the stections of DNA representing the two vectors at twice the standard concentration.andard concentration. Positive and Negative contributionsPositive and Negative contributions

→ → hybridizedhybridized Some single-stranded DNA will be survived.Some single-stranded DNA will be survived.

► Separate the double-stranded DNA from the single stranded Separate the double-stranded DNA from the single stranded DNA of the same lengthDNA of the same length By High-Performance Liquid chromatography(HPLC) purification step.By High-Performance Liquid chromatography(HPLC) purification step. By digesting the double-stranded DNA, using an enzyme.By digesting the double-stranded DNA, using an enzyme.

► Remove the unwanted fragments by HPLC.Remove the unwanted fragments by HPLC.► The individual vectors may be multiplied each by a different The individual vectors may be multiplied each by a different

scalar by adjusting the concentrations.scalar by adjusting the concentrations.

Inner Product of Two VectorsInner Product of Two Vectors

► Obtain three separate samples of each of the two collectionObtain three separate samples of each of the two collections of DNA representing the individual vectors Vs of DNA representing the individual vectors V ii and W and Wii..

► Combine the first pair of samples and measure the rate of Combine the first pair of samples and measure the rate of hybridization, R_, which is proportional to the time rate of inhybridization, R_, which is proportional to the time rate of increase of crease of VV--WW duplex strands representing quantities of opp duplex strands representing quantities of opposite sign.osite sign. The individual contributions to R_ are proportional to the inner produThe individual contributions to R_ are proportional to the inner produ

ct.ct.

m

iiiWV

1

Inner Product of Two VectorsInner Product of Two Vectors

► Incubate separately a Incubate separately a VV and a and a WW sample sample each with DNA polymerase in a suitable each with DNA polymerase in a suitable buffer and the two primersbuffer and the two primers

► The long primer strands grow on the The long primer strands grow on the VV and and WW templates from the 3’ to the 5’ direction, templates from the 3’ to the 5’ direction, producing the producing the complementscomplements to all the to all the VV and and WW strands present. strands present.

'3...'5 12SSSTACACGTTTATCTCGAAATTTATAATGCAAGATE ri

'3...'5 12RRTACRACGTTTATCTCGAAATTTATAATGCAAGATE ri

Inner Product of Two VectorsInner Product of Two Vectors

► Separate the long strands by HPLC to yield Separate the long strands by HPLC to yield the complements the complements VV and and WW..

►Measure the sum of rates of hybridization RMeasure the sum of rates of hybridization R++ of of VV with with WW and and VV with with WW using the third using the third portions of single stranded DNA.portions of single stranded DNA.

► The suitably normalized difference of the The suitably normalized difference of the rates rates RR++ - R_ - R_, each suitably normalized to , each suitably normalized to correct for concentration differences, is the correct for concentration differences, is the inner product of the two vectors.inner product of the two vectors.

Outer Product of Two VectorsOuter Product of Two Vectors

► The outer product matrix The outer product matrix VViiWWjj is formed by joining the singleis formed by joining the single-stranded DNA corresponding to -stranded DNA corresponding to VVii at their 3’ ends to the 5’ at their 3’ ends to the 5’ termini of the termini of the WWjj..

► To ensure that only this type of connection is madeTo ensure that only this type of connection is made the 5’ phosphate residues are removed from the the 5’ phosphate residues are removed from the VVii using for exampl using for exampl

e bacterial alkaline phosphatasee bacterial alkaline phosphatase 5’ termini of the 5’ termini of the WWjj are phosphorylated using for example bacterioph are phosphorylated using for example bacterioph

age T4 polynucleotide kinase.age T4 polynucleotide kinase.► The The WWjj strands are to be further modified by ligating to the strands are to be further modified by ligating to the

3’ termini of the 3’ termini of the WWjj strands a long strand {F} that does not h strands a long strand {F} that does not hybridize significantly to the set of ybridize significantly to the set of EEii’s’s

Outer Product of Two VectorsOuter Product of Two Vectors

► The modified The modified VVii and and WWjj strands are ligated using the four typ strands are ligated using the four types of linker strands to obtain strands of the form {Ees of linker strands to obtain strands of the form {E ii}{E}{Ejj}{F}, }{F}, {{EEii}{E}{Ejj}{F} and so forth.}{F} and so forth.

► The number of ij strands is proportional to the product of thThe number of ij strands is proportional to the product of the concentrations of the e concentrations of the VVii and and WWjj strands and hence to the strands and hence to the desired outer product.desired outer product.

'3......'5 12211 SSSGTATACSSSL rr

'3......'5 12122 SSSGTATACRRRL rr

'3......'5 21214 rr RRGTATACRSSSL

'3......'5 21123 rr RRGTATACRRRRL

ConclusionConclusion

► It is possible to analyze multiplication of It is possible to analyze multiplication of Boolean and real matrices using DNA.Boolean and real matrices using DNA.

► A quantitative calculation can be performed A quantitative calculation can be performed without the necessity of encoding without the necessity of encoding information in the DNA sequence.information in the DNA sequence.

►DNA is used in natural systems for the DNA is used in natural systems for the solution of different types of problemssolution of different types of problems

Thank you for your Thank you for your attention!!!attention!!!