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Coordinate geometry leads into many other topics in school mathematics. The techniques of coordinate geometry are used in calculus, functions, statistics and many other important areas.

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HISTORY

There were three facets of the development of coordinate geometry.

The Greek mathematician Menaechmus (380−320 B C) proved theorems by using a method that was very close to using coordinates and it has sometimes been maintained that he had introduced coordinate geometry.

Apollonius of Perga (262−190 B C) dealt with problems in a manner that may be called an coordinate geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. The results and ideas of the ancient Greeks provided a motivation for the development of coordinate geometry.

Coordinate geometry has traditionally been attributed to René Descartes (1599−1650) and Pierre de Fermat (1601−1665) who independently provided the beginning of the subject as we know it today.

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ANSWERS TO EXERCISES

AB 2 = PQ 2 = UV 2 = ( d b ) 2 + ( c a ) 2

a = 4 or a = 12

Place the triangle so D is at the origin. Then let the coordinates of B and C be (− a , 0) and ( a , 0) respectively. Let the coordinates of A be ( d , c ).

Let P ( x , y ) be a point equidistant from A ( a , b ) and C ( c , d )

PA = PC

2( c a ) x + 2( d b ) y = d 2 + c 2 a 2 b 2

Let the coordinates of the vertices be O (0, 0), A ( a , c ), B ( m , n ) and D ( b , 0)

Show that the midpoint of all the required line segments has coordinates

( m + b + a ), ( n + c )

The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations.

The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations.

© The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE-EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by-nc-nd/3.0/

In addition, we also observed some interesting genes and features in the GIs shared between W1481 and some of the F. nucleatum strains. For example, we found that the horizontally transferred GI3, which was shared by W1481 with the strains of subsp. animalis (11_3_2), subsp. fusiforme (ATCC51190) and subsp. nucleatum (ATCC23726), carries the tripartite ATP-independent (TRAP)-like transporter genes, encoding for TRAP-type C4-dicarboxylate transport system-periplasmic component, TRAP-type C4-dicarboxylate transport system-small permease component and TRAP-type C4-dicarboxylate transport system-large permease component. These TRAP transporters are a high-affinity transport system for the C4-dicarboxylates malate, succinate, and fumarate ( Forward et al. 1997 ). Strikingly, the GI4 which was shared by W1481 with the strains of subsp. polymorphum (ATCC10953 F401) and three strains of subsp. animalis (F0419, D11 7_1), contain a cluster of eight putative CRISPRs genes which may provide the host with acquired and heritable resistance to foreign DNA ( Horvath and Barrangou 2010 ; Marraffini and Sontheimer 2010 ; Jore et al. 2012 ). The distribution of the GIs among the strains indicates that W1481 share a number of GIs with the strains of subsp. animalis , vincentii and polymorphum .

To better understand the pathogenic potential of W1481, we performed a comparative pathogenomic analysis of W1481 along with the strains of the five subspecies of F. nucleatum ( fig. 6 ). W1481 showed the presence of a number of virulence genes which were shared by all or most of the other strains under study mostly related to lipopolysaccharide layer of the bacterial cell surface, such as galE (UDP-glucose 4-epimerase), galU (glucose-1-phosphate uridylyltransferase), gtaB (UTP–glucose-1-phosphate uridylyltransferase), acpXL (acyl carrier protein), kdsA (2-dehydro-3-deoxyphosphooctonate aldolase), lpxA (UDP-N-acetylglucosamine acyltransferase), gmhA/lpcA (phosphoheptose isomerase) and eno (phosphopyruvate hydratase). W1481 also shared the presence of groEL a 60-kDa chaperonin, htpB ( Hsp60 , 60K heat shock protein), hemL (glutamate-1-semialdehyde aminotransferase), tuf (elongation factor Tu) and plr/gapA (glyceraldehyde-3-phosphate dehydrogenase) with the other strains.

F ig . 6.—
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Comparative pathogenomic analysis of W1481. The threshold used in this analysis was 50% sequence identity and 50% sequence completeness.

F ig . 6.—
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Comparative pathogenomic analysis of W1481. The threshold used in this analysis was 50% sequence identity and 50% sequence completeness.

One interesting observation was the presence of ddhABC genes which are related to CDP-glucose pathways. These genes were present in W1481 and only one other strain of F. nucleatum subsp. vincentii ATCC49256. ddhABC genes are involved in lipopolysaccharide O-antigen biosynthesis which have been reported to play a crucial role in effective colonization of host tissues, and evasion of the host immune system by gram negative bacteria ( Lerouge and Vanderleyden 2002 ; Skurnik and Bengoechea 2003 ). None of the other strains under study showed the presence of these genes. W1481 also showed the presence of Putative O-antigen ligase WaaL which is responsible for the ligation of O-antigen onto the core region of the lipid A-core block a crucial step in the lipopolysaccharide biosynthetic pathway ( Han et al. 2012 ).

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Accepted manuscript posted online 16 October 2009 , doi: 10.1128/AEM.00916-09

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Author Affiliations

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ABSTRACT

Fusobacterium nucleatum is a gram-negative oral bacterial species associated with periodontal disease progression. This species is perhaps best known for its ability to adhere to a vast array of other bacteria and eukaryotic cells. Numerous studies of F. nucleatum have examined various coaggregation partners and inhibitors, but it is largely unknown whether these interactions induce a particular genetic response. We tested coaggregation between F. nucleatum ATCC strain 25586 and various species of Streptococcus in the presence of a semidefined growth medium containing saliva. We found that this condition could support efficient coaggregation but, surprisingly, also stimulated a similar degree of autoaggregation. We further characterized the autoaggregation response, since few reports have examined this in F. nucleatum . After screening several common coaggregation inhibitors, we identified l -lysine as a competitive inhibitor of autoaggregation. We performed a microarray analysis of the planktonic versus autoaggregated cells and found nearly 100 genes that were affected after only about 60 min of aggregation. We tested a subset of these genes via real-time reverse transcription-PCR and confirmed the validity of the microarray results. Some of these genes were also found to be inducible in cell pellets created by centrifugation. Based upon these data, it appears that autoaggregation activates a genetic program that may be utilized for growth in a high cell density environment, such as the oral biofilm.

Fusobacterium nucleatum is a gram-negative, anaerobic, oral commensal bacterium that is among the most frequently detected species in human subgingival plaque samples of both healthy and diseased sites ( Cropped Woven Slimleg Pants Pink Victoria Beckham Low Price Cheap Price tCYFA
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). However, during the progression from oral health to periodontitis, the proportion of F. nucleatum increases substantially, which has implicated this species as a contributor to the development of periodontal disease ( Cheap Sale 100% Original Womens Tie Shoulder Jumpsuit Miss Selfridge Cheap Sale Marketable Cheap Sale New Hqrrs
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, 39 ). Currently, the mechanisms of F. nucleatum pathogenicity remain unclear. Butyrate production has been suggested to be a virulence factor ( 2018 Cheap Online Womens Cici Twill Platform Sneakers Opening Ceremony Outlet Low Cost ZLNUVtI8h5
), although clinical data suggest that pathogenicity is strain specific and not inherent among the various F. nucleatum subspecies ( 29 , 33 ). Thus, it has been hypothesized that the association of F. nucleatum with periodontal disease could be due to its role as a “bridge species” for periodontal pathogens ( 23 ).

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/ Chapter
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Affiliations: 1 : Case Western Reserve University, Cleveland, OH 44106-4905
Content Type: Monograph
Publication Year: 2011

Category: Genomics and Bioinformatics

10.1128/9781555817107
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Abstract:

Fusobacterium nucleatum is a gram-negative filamentous anaerobe ubiquitous to the oral cavity. While the focus of this chapter is on F. nucleatum interaction with host cells, the chapter also explains the role of coaggregation in oral microbial community interaction with the host. F. nucleatum is an opportunistic pathogen implicated in various forms of periodontal disease. During periodontal infection, the cell mass of F. nucleatum can increase as much as 10,000-fold, making it one of the most abundant anaerobic species in the diseased sites. In particular, F. nucleatum is one of the leading organisms identified in intrauterine infections causing adverse pregnancy outcomes, including spontaneous miscarriage, preterm birth, and stillbirth. One common feature shared by bacterial pathogens is their ability to adhere to and invade host cells. F. nucleatum modulates an array of host responses upon attachment to and invasion of host cells. A protein complex, FIP, composed of two subunits of 44 and 48 kDa, mediates T-cell suppression. Sequence analysis of TM7a shows that while the majority of the genes are only distantly related to genes found in other organisms, a minority share high sequence similarity with genes found in members of the classes Bacilli , Clostridia , and Fusobacteria , which may result from horizontal gene transfer in the oral cavity. FadA is a unique adhesin in that both the secreted mFadA and the intact pre-FadA are required for function.

Citation: Han Y. 2011. Fusobacterium nucleatum Interaction with Host Cells, p 221-232. Kolenbrander P (ed), . ASM Press, Washington, DC. doi: 10.1128/9781555817107.ch15

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