Article for analog vector algebra computation Allen P. Mils Jr, Bernard Yurke, Philip M Platzman

Introduction ► The chemical operations that can be performed on strands of DNA can be exploited to represent various ordinary algebraic operations including mathematical algorithm. ► Chemical Operations  Ligation  Polymerase  Cutting via restriction enzymes  Base-specific hybridization  Melting of duplex DNA  Destruction of DNA

Introduction ► Objectives  An analog representation for the operations of vector algebra including inner and outer products of dyads and vectors. ► Strategies  Applying Oliver’s work(1997) ► Including a representation for negative real number as well as positive real number represented in Oliver’s work.  Representing the Hopfield associative memory and the feed-forward neural network.

Oliver’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 Work ► ► Reaction sequence used to multiply matrices X and Y. ► ► The desired DNA strands are synthesized such that 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 Work ► ► The square and cube of a matrix and the graph representations of the operations.

Oliver’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.

DNA Vector space ► Vectors  Basis vector and vector space  Concentration [E i ] ► Represented by a DNA sample containing E i strands with concentration [E i ] proportional to the amplitude V i. Basis vector e i 10-dimensional Vector space Amplitude of the i-th component of the vector

DNA Vector space ► Vectors  Practical choice Palindromic restriction enzyme(Bst1107I) recognition sequence Invariant r-mers To assist in hybridization operation A, G, C, T

DNA Vector space ► Vectors  Negative vectors ► Since concentrations are always positive, we need an appropriate representation for negative amplitudes. ► We choose to represent negative unit vectors e i by the sequence of bases complementary to E i. ► As a result, when two vectors are added, any positive and negative amplitude will hybridize and can be removed by digestion using a suitable enzyme or by column separation.

Addition of Vectors ► Combine in one container equal quantities from the two collections of DNA representing the two vectors at twice the standard concentration.  Positive and Negative contributions → hybridized  Some single-stranded DNA will be survived. ► Separate the double-stranded DNA from the single stranded DNA of the same length  By High-Performance Liquid chromatography(HPLC) purification step.  By digesting the double-stranded DNA, using an enzyme. ► Remove the unwanted fragments by HPLC. ► The individual vectors may be multiplied each by a different scalar by adjusting the concentrations.

Inner Product of Two Vectors ► Obtain three separate samples of each of the two collections of DNA representing the individual vectors V i and W i. ► Combine the first pair of samples and measure the rate of hybridization, R_, which is proportional to the time rate of increase of V-W duplex strands representing quantities of opposite sign.  The individual contributions to R_ are proportional to the inner product.

Inner Product of Two Vectors ► Incubate separately a V and a W sample each with DNA polymerase in a suitable buffer and the two primers ► The long primer strands grow on the V and W templates from the 3’ to the 5’ direction, producing the complements to all the V and W strands present.

Inner Product of Two Vectors ► Separate the long strands by HPLC to yield the complements V and W. ► Measure the sum of rates of hybridization R + of V with W and V with W using the third portions of single stranded DNA. ► The suitably normalized difference of the rates R + - R_, each suitably normalized to correct for concentration differences, is the inner product of the two vectors.

Outer Product of Two Vectors ► The outer product matrix V i W j is formed by joining the single-stranded DNA corresponding to V i at their 3’ ends to the 5’ termini of the W j. ► To ensure that only this type of connection is made  the 5’ phosphate residues are removed from the V i using for example bacterial alkaline phosphatase  5’ termini of the W j are phosphorylated using for example bacteriophage T4 polynucleotide kinase. ► The W j strands are to be further modified by ligating to the 3’ termini of the W j strands a long strand {F} that does not hybridize significantly to the set of E i ’s

Outer Product of Two Vectors ► The modified V i and W j strands are ligated using the four types of linker strands to obtain strands of the form {E i }{E j }{F}, {E i }{E j }{F} and so forth. ► The number of ij strands is proportional to the product of the concentrations of the V i and W j strands and hence to the desired outer product.

Conclusion ► It is possible to analyze multiplication of Boolean and real matrices using DNA. ► A quantitative calculation can be performed without the necessity of encoding information in the DNA sequence. ► DNA is used in natural systems for the solution of different types of problems

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