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
Ei  5' AGCTATCGAT3'
Basis vector ei
10-dimensional Vector space
 Concentration [Ei]
► Represented
by a DNA sample containing Ei strands with
concentration [Ei] proportional to the amplitude Vi.
m
V   Vi ei
i 1
Amplitude of the i-th component of the vector
DNA Vector space
► Vectors
 Practical choice
A, G, C, T
Ei  5'TACR1R2 ...Rr N1i N2i N3i ...Nqi S1S2 ...Sr GTA3'
Invariant r-mers
To assist in hybridization operation
Palindromic restriction enzyme(Bst1107I)
recognition sequence
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 ei by the
sequence of bases complementary to Ei.
Ei  5' TAC S r ...S 2 S 1...N q ...N 3 N 2 N 1...R r ...R 2 R1GTA3'
i
► As
i
i
i
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
m
V W
i 1
►
►
i
i
Obtain three separate samples of each of the two
collections of DNA representing the individual vectors Vi
and Wi.
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
Ei  5' AATGCAAGATCGAAATTTATACGTTTATCTTAC S r ...S 2 S 1 3'
Ei  5' AATGCAAGATCGAAATTTATACGTTTATCTTACRr ...R2 R1 3'
► 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 ViWj is formed by joining the
single-stranded DNA corresponding to Vi at their 3’
ends to the 5’ termini of the Wj.
To ensure that only this type of connection is made
 the 5’ phosphate residues are removed from the Vi using for
example bacterial alkaline phosphatase
 5’ termini of the Wj are phosphorylated using for example
bacteriophage T4 polynucleotide kinase.
►
The Wj strands are to be further modified by ligating to
the 3’ termini of the Wj strands a long strand {F} that
does not hybridize significantly to the set of Ei’s
Outer Product of Two Vectors
►
►
The modified Vi and Wj strands are ligated using the
four types of linker strands to obtain strands of the
form {Ei}{Ej}{F}, {Ei}{Ej}{F} and so forth.
The number of ij strands is proportional to the product
of the concentrations of the Vi and Wj strands and
hence to the desired outer product.
L1  5' S1S2 ...Sr GTATACS r ...S 2 S13'
L2  5' Rr ...R2 R1GTATACS r ...S 2 S 1 3'
L3  5' R r ...R 2 R1GTATACR1R2 ...Rr 3'
L4  5' S1S2 ...Sr GTATACR1R2 ...Rr 3'
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
Thank you for your
attention!!!