Attempts to routinely sequence a human genome by increasing Published online 9 October ; doi Instead, the sequencing equivalent of the transistor and densities. Additionally, the physical separation obtained by the discrete yet successful, provided the basis for miniaturization and scaling. Two wells of a microtiter plate can be replaced by tightly packed reactions on general approaches had been proposed. On the other were technologies that sequenced by detecting pyrophosphate release with an enzymatic cascade ending A new way to conduct experiments in parallel in luciferase and detection of the emitted light.
The latter approach was In developing a system based on a high-throughput and highly parallel chosen for the platform because direct incorporation of natural format, one of the key problems was how to supply reagents to the large nucleotides seemed more efficient than repeated cycles of incorporation, quantities of simultaneous reactions required for projects on the scale detection and cleavage.
This premise has been shown to be correct, with of human genome sequencing. This presents a challenge because each competing next-generation technologies facing a current limit between reaction requires a recurring source of fresh reagents with each base 25 and 50 bases ref. Again the importance of creating a parallel and Pyrosequencing has been available to the scientific community since highly dense process became essential to enable the technique to achieve the mids as a genotyping tool, but was not considered powerful the necessary scale.
The solution to this problem was simple and elegant: enough for standard sequencing needs because of the short read-lengths at high reaction densities, diffusion from a laminar flow stream itself is it generated, relegating it to the role of single-nucleotide polymorphism sufficient to bring fresh reagents to each reaction and to wash both by- SNP -based genotyping. At that time, the technology was being used in products and unincorporated reagents away.
This laminar flow concept microtiter plates to process up to 96 genotypes in a sequential fashion28, has been adopted by each subsequent next-generation platform. To separate the individual reactions, early simulations of reactions on Pyrosequencing had not yet shown the capability for de novo sequenc- a flat surface showed that the densities would be limited to hundreds to ing—which requires accurate de novo identification of nucleotides at thousands of reactions per cm2.
However, the goal was to sequence tens every incorporation rather than the simple confirmation of bases at or hundreds of thousands of reactions per cm2 to conserve the expensive known positions as used in SNP-based genotyping—and longer read- reagents that would need to flow over the reactions and enable efficient lengths beyond the few bases required for genotyping Higher densities were accomplished by placing the reactions However, because pyrosequencing is based on the detection of light in wells, with the well depth further isolating the individual reactions produced whenever a nucleotide is incorporated Fig.
Although this approach greatly reduced the area needed require a physical separation process like electrophoresis to resolve the to carry out the reactions, the sheer number of desired reactions still next base in the DNA strand. Slides with high-density wells were made in two steps. These wells would serve as individual reactors to allow individual enzymatic reactions to take PPi place in each of the separate wells.
The template strand is light from the reaction to be captured by placing the slide on top of a represented in red, the annealed primer is shown in black and the DNA polymerase is shown as the green oval.
The wells also would facilitate the addition of the other converted to ATP by the sulfurylase blue arrow. Luciferase red arrow uses reagents needed for the light-producing reactions. Instead of having the ATP to convert luciferin to oxyluciferin, producing light. This immobilization step ensured that the reaction was local to the well where the light would be captured by the fiber-optic bundle and also reduced costs, as enzyme replenishment would not be necessary under flow conditions.
Optimizing the sequencing process alone was insufficient to enable truly high-throughput genome sequencing. These c steps were followed by time-consuming and expensive amplification of the individual clones, and cleanup of templates—all before a single i sequence reaction could be run.
This would provide a complete sequencing process covering iii all aspects from genome of interest to finished sequence that would be A T G C ii completely in vitro, massively parallel and scalable as improvements to density and read-length were achieved on the fiber-optic well plate Figure 2 Overview of the sequencing technology. The goals of developing the solid-phase sequenc- optic slide part of object iii , and a computer that provides the necessary ing methods, and optimizing read-lengths in the microwells were closely user interface and instrument control part of object iii.
As reac- tants quickly diffused from the wells, the synthesis cycle could be quickly fluidics, surface chemistry and enzymology Leamon and Rothberg34 , repeated. Instead of removing or degrading unincorporated nucleotides including identifying superior polymerases, optimizing the sequenc- with apyrase27, long read-lengths — base-reads were obtained ing reaction at higher temperatures, and replacing and rebalancing the by the rapid diffusion of unincorporated nucleotides and reaction by- components in the enzyme cascade Fig.
Along with the efficient removal of unincorporated bases, tigated, but not commercialized. These included the use of reversible the efficiency of polymerase extensions in each cycle was greatly facili- terminators to increase accuracy across homopolymers, strategies for tated by the fast diffusion and removal of resid- ual concentrations of inorganic phosphates.
Table 1 Applications introduced through collaborations with Complete polymerase extensions enabled Subsequent acquisition of long, accurate sequencing reads publication However, because of the difficulty of maintaining stable the same template and alternative enzyme-immobilization methods Although emulsion-based techniques for clonal production of tem- Template preparation.
A completely in vitro and massively parallel tem- plates without the need for bacteria were of interest from the start, suit- plate preparation method was needed to provide low cost templates able surfactants were not known at the time to allow the emulsions for high-throughput sequencing.
For this reason isothermal approaches were explored. Although amplification yields from rolling circle amplification RCA reactions were Box 1 Next-generation sequencing formats extremely high able to generate solid masses As of September , three additional next-generation technologies have come to of amplified DNA in the wells , the majority market.
As with the RCA experiments, early amplification of template material, thereby enabling true single-molecule sequencing. These systems are described in fiber-optic slide.
Amplification was achieved more detail in Table 2. This binding site studies. These technologies will also be applicable in cases in which short method was successful, but inefficient: the sequences, such as miRNAs, are probed. Contamination colleagues Nonetheless, the process was able colleagues These results suggest that for many applications the Solexa and SOLiD to amplify templates from a whole-genome platforms might require fourfold more sequencing data to achieve genome coverage library for subsequent sequencing resulting comparable to that derived using long-read technologies.
In addition, identification of in the first de novo sequencing of any genome protein-coding regions in infectious disease and other metagenomic projects usually using a non-Sanger, non-Gilbert sequencing requires longer reads derived from traditional Sanger or Sequencers. One area method, as well as the first sequencing of a that does show promise for the short-read technologies is resequencing of exons after genome adenovirus with complete in vitro enrichment85 and the identification of chromosomal rearrangements However, all novel sample preparation GenBank AY The finally obtained with the incorporation of relative costs per base and read-lengths provided by the SOLiD, Solexa and platforms, surfactants used in the manufacture of explo- and the segmentation of the next-generation sequencing market based on those factors is sives, where thermostable segregation of die- shown in Figure 3.
It will be interesting to monitor these technologies over time as they attempt to The success of this formulation dramatically increase read-length and throughput while reducing cost per base. The emPCR system proved highly system.
It will be interesting to Along with the increases in sequencing qual- see if the density of the system can continue to increase by use of smaller beads, ity and read-length, continued improvements while the tenfold advantage in read-length and time per base extension is retained. The library preparation methodology was fairly well determined from the conception of the project, with the generation of a randomly fragmented library from a sample genome followed by Coun de n ting a ovo s pplic equenc ligation of distinct adapters to opposing ends of the template for subse- atio ons ing quent amplification.
Removal of — — the bacterial cloning step dramatically increased the speed and efficiency 0 0 50 of the method while sidestepping the potential loss of sequence coverage due to bacteria-induced bias. This methodology would also Average read-length in bases prove to be ideal for metagenomic analysis and ancient DNA studies. A paired-end library preparation method was developed to enable the de Figure 3 Next-generation market segmentation. Sequencing application novo assembly of complex genomes, span repetitive regions and allow segmentation as a function of cost per megabase, and read-length.
Data for the systematic study of genome structure, including duplications and costs per run and read-lengths for Solexa and SOLiD were obtained from a variety of sources, including respective company websites and a recent other rearrangements The development of this paired-end library technology review Note the run times for data points are 5, 8 and 10 h, respectively, and those Applications of the technology for Solexa and SOLiD are 72 h.
With the realization that the greatest scientific impact would be made by demonstrating the utility of sequencing in as wide a range as applications as possible Table 1 , set up collaborations in which the company would sequence and analyze samples with both industrial and academic determination of the pathogenic content of the bacteria responsible for researchers.
These collaborations were further designed to demonstrate pneumonia, meningitis and urinary tract infections Additionally, recent human sequencing with the ability to culture or clone the DNA from the sample of interest. To demonstrate the and missing from the reference sequence Rothberg and colleagues Rresistant M.
The explosion of interest in small RNAs, including tis 6 Mb strain. This project clearly illustrated the advantages of microRNAs miRNAs , in was perfectly timed with the commer- sequencing both in terms of speed and accuracy; using traditional Sanger cial availability of the sequencing system and the ability to sequence technologies, sequencing one 4-Mb genome and three 6-Mb genomes hundreds of thousands of templates simultaneously.
The system also avoided possible biases introduced by the cesses, as shown by the following examples. An early pivotal study was traditional cloning process, generating high-quality data that permitted a collaborative effort in which miRNAs in Arabidopsis thaliana were accurate identification of the two missense mutations that conferred investigated This was closely followed by another collabora- R resistance The M. These studies paved the way for also underscored the value of the Sequencer for bacterial sequenc- additional studies on small RNAs in human, chimpanzee56, zebrafish57 ing applications, leading to comparative genomic studies45, de novo and tumor cell lines Template DNA is nebulized and size-selected to 8-h run Leamon, Rothberg and colleagues16 , a subsequent product produce a population of double-stranded fragments ranging from release generated an average of megabases of base-reads.
Two distinct oligonucleotide adapters are ligated onto the With current work using higher density fiber-optic plates, base-reads fragments, providing priming sites for subsequent amplification and in excess of are expected to generate between and sequencing. One of the adapters is biotinylated, permitting collection of megabases per run.
The relatively low throughput per sequencing single-stranded templates. Template-covered DNA capture beads are loaded into individual wells etched into the surface of a fiber-optic slide. The sequencing process uses an enzymatic cascade to generate light from inorganic pyrophosphate PPi molecules released by the incorporation of nucleotides as a polymerase replicates the template DNA Leamon, Rothberg and colleagues Individual nucleotides are provided to the open wells by flowing them over the fiber-optic slide.
The number of photons generated by the cascade is proportional to the number of nucle- otides incorporated by the polymerase and the release of the PPi gener- ated by the individual sequencing reactions. Genome Analyzer The Solexa system operates via a sequencing-by-synthesis process Sample density on the Solexa platform is currently million samples Illumina that incorporates fluorescently labeled nucleotides into immobilized per cm2 with a sequencing output of 1 gigabase, higher than the template strands.
Amplification is conducted in situ via bridge ampli- system per run but comparable in terms of total bases per hour. For example, any two distinct objects will be distinguished by the context 'x MM', where M is a set which contains one but not the other.
Frege himself might defend against this by insisting that 'xM M' is not a primitive concept and must be replaced by the concept F x of which M is the extension.
I have already expressed the view that it would have been entirely unreasonable for Frege to have supposed Cantor to have rejected this trivial form of Leibniz's principle or to have overlooked it in formulating his doctrine of cardinal numbers as sets of pure units.
So Frege's objection would then be that there is no such concept F x , not already involving the identity relation, which distinguishes between two 'pure units' of the cardinal-set. Frege p. Certainly no point in Euclidean space is distinguished from any other by a concept, unless that concept itself is defined by reference to specific points, since the space is homogeneous.
But there is a difficulty with individuating points by means of concepts which themselves refer to points. For example, the points p and q may be distinguished by the concept 'x is between r and s', which is satisfied by p and not by q. But then q satisfies a corresponding concept 'x lies between t and u, e. The problem is that our grounds for calling these two concepts distinct is only that r,s and t,u are distinct i. Hence, there is a circle. But it would be hard to accept the non-trivial form of Leibniz's principle as a metaphysical principle based upon such a belief.
We must have identity - hence the 1; but we must have difference - hence the indices; only unfortunately, the latter undo the work of the former. Frege and Jevons collaborate in a confusion here. If the different occurrences of '1' do not 29 Leibniz did not face this difficulty. For him, the principle of identity of indiscernibles applies to substances which have no real relations , and not to ideal things such as points in space.
But that is because he thinks that, if cardinals are sets, then their addition is just their union. He is right that one cannot take the units in 5 to be the unit in 1, since there are five of the former and only one of the latter.
Frege's argument, were it valid, would not simply be an argument against cardinals being sets of pure units, it would be an argument against the cardinal of a set M being a set equipollent to M at all.
I am persuaded by Dedekind's grammatical argument that numbers are not sets; but Frege's line of argument would exclude even the representation of cardinals by initial von Neumann ordinals.
Of course, Frege might have made the argument that, since the identification of cardinals with sets does not admit the identification of cardinal addition with set-theoretic union, then there is no point in regarding cardinals as sets at all.
But this is not the argument that he gave. IX One problem with reading the literature on the number concept prior to Cantor, Dedekind and Frege is that, aside from Bolzano, the authors generally have not fully distinguished the notion of a set and tended to subsume it under a more general notion of a 'multitude' or 'plurality'. Anything with proper parts was regarded as a plurality - a line segment, Socrates, a heap of stones, a flock of sheep.
It was understood from the time of Plato that number does not unambiguously apply to pluralities. Socrates is one but has many parts, the flock of ten sheep also includes a plurality of twenty sheeps' eyes and a plurality of forty sheeps' legs, etc.
Aristotle explicitly understood that assigning number to pluralities in this sense requires a prior choice of the parts to be numbered. There was also a surviving conceptual difficulty with unit sets, which is reflected in the sometime rejection of the number 1 by the classical Greeks: if what can be numbered is an object X relative to a choice of unit, the unit set of X would be X qua part of X, which was indistinguishable from X.
He writes Some writers define the number as a set or multitude or plurality. All of these views suffer from the drawback that the concept will not then cover the numbers 0 and 1. Moreover, these terms are utterly vague: sometimes they approximate in meaning to "heap" or "group" or "aggregate" "A g g r e g a t " , referring to a juxtaposition in space, sometimes they are so used as to be practically equivalent to "number", only vaguer.
In Paradoxien des Unendlichen , Bolzano had already made a distinction between what he called an 'Inbegriff' and a set: There exist aggregates I n b e g r i f f e which agree in containing the selfsame members, and nevertheless present themselves as d i f f e r e n t when seen under different aspects or under different conceptions, and this kind of difference we call 'essential'.
For example: an unbroken tumbler and a tumbler broken into pieces, considered as a drinking vessel. We call the ground of distinction between two such aggregates their mode of combination or their a r r a n g e m e n t. Also see the second quote below from Bolzano's Paradoxien des Unendlichen. Notice also that Cantor did not clearly distinguish between Bolzano's notions of Inbegriff and Menge in the quote above from , p.
A further difficulty with reading the pre-twentieth century literature on the number concept is manifested in Frege's "[these terms] are so used as to be practically equivalent to 'number', only vaguer". The fact is that the term "number", in the sense of the whole numbers, often really did just mean a finite set - in the somewhat confused sense of 'set' that we have just discussed.
So a 'number' is given only relative to the choice of unit. A unit is that with respect to which each of the things that exist is called one. A number is a multitude composed of units. In Definition 1 Euclid says that numbering begins with the choice of unit, of what is to be counted - what is to be called one.
It is true that 'unit' in Definition 1 refers to a property and in Definition 2 to objects having that property. But that is a frivolous objection. We use the word 'man' sometimes to refer to a property and sometimes to men. So things may be called 'one' after the choice of unit in the same way that men may be called 'man'. It is because Frege does not understand that 'one' is a c o m m o n name, applying to all units once the unit has been chosen , that he misunderstands Euclid.
But he can't have it both ways: if he believes that the term 'set' can only refer to heaps and the like, then he should also agree that number can be assigned only after the choice of unit. Hume also uses the term 'number' to mean a set. Frege goes on to say that the definition of numerical equality in terms of one-to-one correspondence raises certain "logical doubts and difficulties.
From which it seems to follow that we ought not to define it specially for the case of numnbers. I , that by 'number' Hume is referring to finite sets and that, when he speaks of equality of numbers, he is not referring to the identity relation but to the relation of equinumerosity, w h i c h indeed is to be defined specially for the case of 'numbers', i.
The 'logical doubts and difficulties' were created by Frege's incorrect reading, not by Hume's conception. Bolzano also used the term 'number' for sets. Certainly the term 'number' did not always refer to a set. Thus people spoke of 'the number ten' and of the set and number of prime numbers less than n, etc.
Into the nineteenth century, every theorem of number theory could be understood as a statement about an arbitrary finite set, free from any assumption of the existence of an infinite set.
Moreover, there was always the resource exploited in analytic number theory of regarding the natural numbers as embedded in the system of real numbers. The move towards treating the natural numbers as forming an autonomous system of objects dates from later in the century.
It required the explicit admission of the actual infinite into mathematics, and the motivation would seem to be the arithmetical foundation of the system of real numbers. Rational numbers are constructed from natural numbers and real numbers are, for example, sets of rationals. If the natural numbers are not to form an autonomous system of objects, then it is hard to make sense of this construction.
We have already quoted his attribution to Hume of the definition of this concept as meaning equipollence. Dummett writes "By the time that Frege wrote Grundlagen, the definition had already become a piece of mathematical orthodoxy, though Frege undoubtedly gave it its most exact formulation and its most acute philosophical defense. We have already noted his misunderstanding of Hume. But there is another and more compelling respect in which his citation is misleading and, in particular, slights Cantor's contribution.
I do not mean the fact that Frege refers to Cantor's paper rather than to his earliest 33 Dummett's explanation pp. In other words, we need a set i.
Frege and Cantor, on the other hand were concerned with the general notion of cardinal. The extension to this general notion was not a trivial matter. As late as , in his monograph just cited, Bolzano, who did understand the notion of set to include infinite sets, had argued that the characterization of numerical equality in terms of one-to-one correspondence, though correct for finite sets, could not be applied to infinite sets.
His reason was the traditional obstacle to a coherent theory of infinite numbers: It would then happen that infinite sets could be numerically equal to proper subsets of themselves. XI One must wonder why Dummett wrote that Frege gave the definition of equinumerosity in terms of one-to-one correspondence of sets "its most exact formulation and its most acute philosophical defense".
But anyway it is stated with admirable clarity by both Bolzano and Cantor, even though the former did not accept it as the definition of equinumerosity in the case of infinite sets. As for a defense, who was attacking it and why was defense needed, philosophical or otherwise?
Certainly one attack that needed to be answered was Bolzano's, which we have already mentioned. Cantor responded to this and Frege made no mention of it at all. In this paper, Cantor does not define the general concept of equipollence, although he does produce the first significant result concerning infinite powers. On the ordinal conception, it is not abstract sets but well-ordered sets to which number applies.
Again, this analysis was undertaken by Cantor, not by Frege. But if we try to do it in the other way, by putting together identicals Gleichem , the result runs perpetually together into one and we never reach a plurality. But distinguish two questions which, in his discussion of earlier authors, Frege tended to confuse: What are the things to which number applies? And, what are numbers? The first horn of his dilemma concerns the first question. And the things to be numbered are not 'agglomerations' but sets, which indeed arise by 'putting together' different distinct objects.
These sets w e r e called 'numbers' by some of these authors, and this is one source of Frege's confusion. The second horn of Frege's dilemma can concern only the view that the numbers themselves are sets of pure units. The argument is that, if the units are not distinguished by their properties, then they will be identical.
We have already discussed this view and concluded that the notion of cardinal number as a set of pure units, though unattractive, is by no means incoherent. Distinct units are indeed distinguished by their properties; but when from a set of two cats, one white and one black, we 'abstract' the number two as a set of pure units, the units are not white and black, respectively, and they are not cats.
But also, unlike Frege, I find no sign of the conception of number as set of pure units in the quotes from other authors that he offers in in contrast to Cantor in his later writings and to Husserl's book. His reading seems to me to have been misdirected by two related things: his interpretation of "gleich" to mean identity and his failure to understand the historical use of the term "number" to mean what is numbered.
He means rather that they are being identified a s the things to be counted. If we put together Thomae's "abstract …set of things" with Lipschitz's "the concept of the number…", then it seems that we are abstracting from the particular nature of the elements of a set t o obtain the number of the set. But if we read only what Lipschitz wrote, then we are, in considering separate things, disregarding those characteristics which serve to distinguish them and are left with the concept of the number anzahl of the things in question.
But by "anzahl" here Lipschitz means the set. So Frege is in complete agreement with Lipschitz: The set or concept whose extension is the set is obtained by abstracting from the differences among the elements of the set. The citation of Thomae is equally misleading. His ultimate concern is with the theory of analytic functions and so with the complex numbers. He is sketching the construction of these numbers from the natural numbers. His account is 'formalistic', in the sense that he treats the natural numbers as signs and the real numbers as infinite sequences of signs.
This account is by no means as defective as Frege makes it out to be e. The numbers, being for Thomae signs, are certainly not sets of pure units. The context of the above quote from Thomae is this: 37 And Frege's treatment of the 'formalism' of Heine in is totally unjustified. Each individual of the set is called a unit, and, as a consequence of the required abstraction from all distinctive peculiarities, one may replace any unit by any other.
The units are equal to one another. The sense in which they are equal is explained: one can be substituted for any other without altering the name assigned, i.
This is consistent with his view that Cantor's notion of a set can only be understood as the extension of a concept. Indeed, we saw that this was true of Euclid. But Frege seems to have thought that this was a source of confusion, whereas one would think that it would have been a commonplace observation: "Man" is sometimes used to denote a particular man and sometimes to denote the property of being a man.
The difference between 'man' and 'unit' in this respect is that the meaning of the former is fixed and the meaning of the latter is relative and must be specified in any context - as meaning 'man in the room' in one context, 'sheep in the flock' in another, etc.
XII Ultimately, Frege's contribution with respect to the definition of equinumerosity was to replace Cantor's sets as the objects of number attributions by concepts. Indeed, his proposal in his review of Cantor is that we should understand the term 'set' to refer to extensions of concepts. Perhaps it is this idea that Dummett thinks renders the definition more precise. In the case of finite numbers, Frege's proposal would indeed have clarified the discussion of number among his contemporaries: the meaning of choosing a unit and of all units being equal is well analyzed in terms of choosing a concept, in something like Frege's sense of concept.
The number of people in the room is an attribute of the concept 'x is in the room', where x ranges over people. But notice that, in the above example, we do not really have Frege's notion of concept, since x ranges over people and, for Frege, the variable ranges over all objects. In the example given, the appropriate concept for Frege would be 'x is a person in the room'. But, when we admit Fregean concepts in general, that is concepts of the form F x , where x ranges over 'all objects', then Frege's idea that number is attributable to concepts goes wrong.
In the case of infinite numbers, the fact is that Cantor had already noted in his that there are concepts, for example the concept that x is an ordinal or cardinal , which do not have a power. The file will be sent to your Kindle account. It may takes up to minutes before you received it. Please note : you need to verify every book you want to send to your Kindle.
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