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64-614: The WPGMA method is similar to its unweighted variant, the UPGMA method. The WPGMA algorithm constructs a rooted tree ( dendrogram ) that reflects the structure present in a pairwise distance matrix (or a similarity matrix ). At each step, the nearest two clusters, say i {\displaystyle i} and j {\displaystyle j} , are combined into a higher-level cluster i ∪ j {\displaystyle i\cup j} . Then, its distance to another cluster k {\displaystyle k}

128-403: A {\displaystyle a} and b {\displaystyle b} . Let u {\displaystyle u} denote the node to which a {\displaystyle a} and b {\displaystyle b} are now connected. Setting δ ( a , u ) = δ ( b , u ) = D 1 (

192-403: A {\displaystyle a} and b {\displaystyle b} . Let u {\displaystyle u} denote the node to which a {\displaystyle a} and b {\displaystyle b} are now connected. Setting δ ( a , u ) = δ ( b , u ) = D 1 (

256-530: A {\displaystyle a} or b {\displaystyle b} to v {\displaystyle v} , and e {\displaystyle e} to v {\displaystyle v} are equal and have the following length: δ ( a , v ) = δ ( b , v ) = δ ( e , v ) = 22 / 2 = 11 {\displaystyle \delta (a,v)=\delta (b,v)=\delta (e,v)=22/2=11} We deduce

320-530: A {\displaystyle a} or b {\displaystyle b} to v {\displaystyle v} , and e {\displaystyle e} to v {\displaystyle v} are equal and have the following length: δ ( a , v ) = δ ( b , v ) = δ ( e , v ) = 22 / 2 = 11 {\displaystyle \delta (a,v)=\delta (b,v)=\delta (e,v)=22/2=11} We deduce

384-402: A {\displaystyle a} ), Bacillus stearothermophilus ( b {\displaystyle b} ), Lactobacillus viridescens ( c {\displaystyle c} ), Acholeplasma modicum ( d {\displaystyle d} ), and Micrococcus luteus ( e {\displaystyle e} ). Let us assume that we have five elements (

448-391: A , b ) {\displaystyle (a,b)} and element e {\displaystyle e} . Let v {\displaystyle v} denote the node to which ( a , b ) {\displaystyle (a,b)} and e {\displaystyle e} are now connected. Because of the ultrametricity constraint, the branches joining

512-391: A , b ) {\displaystyle (a,b)} and element e {\displaystyle e} . Let v {\displaystyle v} denote the node to which ( a , b ) {\displaystyle (a,b)} and e {\displaystyle e} are now connected. Because of the ultrametricity constraint, the branches joining

576-570: A , b ) / 2 {\displaystyle \delta (a,u)=\delta (b,u)=D_{1}(a,b)/2} ensures that elements a {\displaystyle a} and b {\displaystyle b} are equidistant from u {\displaystyle u} . This corresponds to the expectation of the ultrametricity hypothesis. The branches joining a {\displaystyle a} and b {\displaystyle b} to u {\displaystyle u} then have lengths δ (

640-570: A , b ) / 2 {\displaystyle \delta (a,u)=\delta (b,u)=D_{1}(a,b)/2} ensures that elements a {\displaystyle a} and b {\displaystyle b} are equidistant from u {\displaystyle u} . This corresponds to the expectation of the ultrametricity hypothesis. The branches joining a {\displaystyle a} and b {\displaystyle b} to u {\displaystyle u} then have lengths δ (

704-411: A , b ) , c ) = ( D 1 ( a , c ) × 1 + D 1 ( b , c ) × 1 ) / ( 1 + 1 ) = ( 21 + 30 ) / 2 = 25.5 {\displaystyle D_{2}((a,b),c)=(D_{1}(a,c)\times 1+D_{1}(b,c)\times 1)/(1+1)=(21+30)/2=25.5} D 2 ( (

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768-403: A , b ) , c ) = ( D 1 ( a , c ) + D 1 ( b , c ) ) / 2 = ( 21 + 30 ) / 2 = 25.5 {\displaystyle D_{2}((a,b),c)=(D_{1}(a,c)+D_{1}(b,c))/2=(21+30)/2=25.5} D 2 ( ( a , b ) , d ) = ( D 1 (

832-403: A , b ) , d ) = ( D 1 ( a , d ) + D 1 ( b , d ) ) / 2 = ( 31 + 34 ) / 2 = 32.5 {\displaystyle D_{2}((a,b),d)=(D_{1}(a,d)+D_{1}(b,d))/2=(31+34)/2=32.5} D 2 ( ( a , b ) , e ) = ( D 1 (

896-494: A , b ) , e ) {\displaystyle ((a,b),e)} and ( c , d ) {\displaystyle (c,d)} to r {\displaystyle r} then have lengths: δ ( ( ( a , b ) , e ) , r ) = δ ( ( c , d ) , r ) = 33 / 2 = 16.5 {\displaystyle \delta (((a,b),e),r)=\delta ((c,d),r)=33/2=16.5} We deduce

960-494: A , b ) , e ) {\displaystyle ((a,b),e)} and ( c , d ) {\displaystyle (c,d)} to r {\displaystyle r} then have lengths: δ ( ( ( a , b ) , e ) , r ) = δ ( ( c , d ) , r ) = 35 / 2 = 17.5 {\displaystyle \delta (((a,b),e),r)=\delta ((c,d),r)=35/2=17.5} We deduce

1024-457: A , b ) , e ) {\displaystyle ((a,b),e)} and ( c , d ) {\displaystyle (c,d)} . Let r {\displaystyle r} denote the (root) node to which ( ( a , b ) , e ) {\displaystyle ((a,b),e)} and ( c , d ) {\displaystyle (c,d)} are now connected. The branches joining ( (

1088-457: A , b ) , e ) {\displaystyle ((a,b),e)} and ( c , d ) {\displaystyle (c,d)} . Let r {\displaystyle r} denote the (root) node to which ( ( a , b ) , e ) {\displaystyle ((a,b),e)} and ( c , d ) {\displaystyle (c,d)} are now connected. The branches joining ( (

1152-563: A , b ) , e ) ) × 1 + D 3 ( d , ( ( a , b ) , e ) ) × 1 ) / ( 1 + 1 ) = ( 30 × 1 + 36 × 1 ) / 2 = 33 {\displaystyle D_{4}((c,d),((a,b),e))=(D_{3}(c,((a,b),e))\times 1+D_{3}(d,((a,b),e))\times 1)/(1+1)=(30\times 1+36\times 1)/2=33} The final D 4 {\displaystyle D_{4}} matrix is: So we join clusters ( (

1216-425: A , b ) , e ) , c ) = ( D 2 ( ( a , b ) , c ) + D 2 ( e , c ) ) / 2 = ( 25.5 + 39 ) / 2 = 32.25 {\displaystyle D_{3}(((a,b),e),c)=(D_{2}((a,b),c)+D_{2}(e,c))/2=(25.5+39)/2=32.25} Of note, this average calculation of the new distance does not account for

1280-457: A , b , c , d , e ) {\displaystyle (a,b,c,d,e)} and the following matrix D 1 {\displaystyle D_{1}} of pairwise distances between them : In this example, D 1 ( a , b ) = 17 {\displaystyle D_{1}(a,b)=17} is the smallest value of D 1 {\displaystyle D_{1}} , so we join elements

1344-457: A , b , c , d , e ) {\displaystyle (a,b,c,d,e)} and the following matrix D 1 {\displaystyle D_{1}} of pairwise distances between them : In this example, D 1 ( a , b ) = 17 {\displaystyle D_{1}(a,b)=17} is the smallest value of D 1 {\displaystyle D_{1}} , so we join elements

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1408-670: A , d ) + D 1 ( b , d ) ) / 2 = ( 31 + 34 ) / 2 = 32.5 {\displaystyle D_{2}((a,b),d)=(D_{1}(a,d)+D_{1}(b,d))/2=(31+34)/2=32.5} D 2 ( ( a , b ) , e ) = ( D 1 ( a , e ) + D 1 ( b , e ) ) / 2 = ( 23 + 21 ) / 2 = 22 {\displaystyle D_{2}((a,b),e)=(D_{1}(a,e)+D_{1}(b,e))/2=(23+21)/2=22} Italicized values in D 2 {\displaystyle D_{2}} are not affected by

1472-414: A , e ) + D 1 ( b , e ) ) / 2 = ( 23 + 21 ) / 2 = 22 {\displaystyle D_{2}((a,b),e)=(D_{1}(a,e)+D_{1}(b,e))/2=(23+21)/2=22} Italicized values in D 2 {\displaystyle D_{2}} are not affected by the matrix update as they correspond to distances between elements not involved in

1536-468: A , u ) = δ ( b , u ) = 17 / 2 = 8.5 {\displaystyle \delta (a,u)=\delta (b,u)=17/2=8.5} ( see the final dendrogram ) We then proceed to update the initial distance matrix D 1 {\displaystyle D_{1}} into a new distance matrix D 2 {\displaystyle D_{2}} (see below), reduced in size by one row and one column because of

1600-468: A , u ) = δ ( b , u ) = 17 / 2 = 8.5 {\displaystyle \delta (a,u)=\delta (b,u)=17/2=8.5} ( see the final dendrogram ) We then proceed to update the initial distance matrix D 1 {\displaystyle D_{1}} into a new distance matrix D 2 {\displaystyle D_{2}} (see below), reduced in size by one row and one column because of

1664-423: A helicase -like activity, making it able to unwind DNA strands. Its optimum functional temperature is between 60 and 65 °C and it is denatured at temperatures above 70 °C. These features make it useful in loop-mediated isothermal amplification (LAMP) . LAMP is similar to the polymerase chain reaction (PCR) but does not require the high temperature (96 °C) step required to denature DNA. In 2013,

1728-554: A thermostable group II intron reverse transcriptase (TGIRT), GsI-IIC-MRF, from G. stearothermophilus was found to retain activity up to 70 °C and to exhibit high processivity and a low error rate. These properties make this enzyme useful for reverse transcribing long and/or highly structured RNA molecules. A method for determining RNA secondary structure , DMS-MaPseq, uses this enzyme because it converts normal RNA to DNA accurately but introduces mutations at unpaired bases that have been methylated by dimethyl sulfate , and

1792-400: A constant-rate assumption - that is, it assumes an ultrametric tree in which the distances from the root to every branch tip are equal. When the tips are molecular data ( i.e. , DNA , RNA and protein ) sampled at the same time, the ultrametricity assumption becomes equivalent to assuming a molecular clock . This working example is based on a JC69 genetic distance matrix computed from

1856-415: A constant-rate assumption: it produces an ultrametric tree in which the distances from the root to every branch tip are equal. This ultrametricity assumption is called the molecular clock when the tips involve DNA , RNA and protein data. This working example is based on a JC69 genetic distance matrix computed from the 5S ribosomal RNA sequence alignment of five bacteria: Bacillus subtilis (

1920-472: A drawback of the alternative single linkage clustering method - the so-called chaining phenomenon , where clusters formed via single linkage clustering may be forced together due to single elements being close to each other, even though many of the elements in each cluster may be very distant to each other. Complete linkage tends to find compact clusters of approximately equal diameters. UPGMA UPGMA ( unweighted pair group method with arithmetic mean )

1984-423: A drawback of the alternative single linkage clustering method - the so-called chaining phenomenon , where clusters formed via single linkage clustering may be forced together due to single elements being close to each other, even though many of the elements in each cluster may be very distant to each other. Complete linkage tends to find compact clusters of approximately equal diameters. A trivial implementation of

WPGMA - Misplaced Pages Continue

2048-533: Is a rod-shaped, Gram-positive bacterium and a member of the phylum Bacillota . The bacterium is a thermophile and is widely distributed in soil, hot springs, ocean sediment, and is a cause of spoilage in food products. It will grow within a temperature range of 30–75 °C. Some strains are capable of oxidizing carbon monoxide aerobically. It is commonly used as a challenge organism for sterilization validation studies and periodic check of sterilization cycles. The biological indicator contains spores of

2112-461: Is a simple agglomerative (bottom-up) hierarchical clustering method. It also has a weighted variant, WPGMA , and they are generally attributed to Sokal and Michener . Note that the unweighted term indicates that all distances contribute equally to each average that is computed and does not refer to the math by which it is achieved. Thus the simple averaging in WPGMA produces a weighted result and

2176-574: Is now complete. It is ultrametric because all tips ( a {\displaystyle a} to e {\displaystyle e} ) are equidistant from r {\displaystyle r}  : δ ( a , r ) = δ ( b , r ) = δ ( e , r ) = δ ( c , r ) = δ ( d , r ) = 16.5 {\displaystyle \delta (a,r)=\delta (b,r)=\delta (e,r)=\delta (c,r)=\delta (d,r)=16.5} The dendrogram

2240-573: Is now complete. It is ultrametric because all tips ( a {\displaystyle a} to e {\displaystyle e} ) are equidistant from r {\displaystyle r}  : δ ( a , r ) = δ ( b , r ) = δ ( e , r ) = δ ( c , r ) = δ ( d , r ) = 17.5 {\displaystyle \delta (a,r)=\delta (b,r)=\delta (e,r)=\delta (c,r)=\delta (d,r)=17.5} The dendrogram

2304-564: Is simply the arithmetic mean of the average distances between members of k {\displaystyle k} and i {\displaystyle i} and k {\displaystyle k} and j {\displaystyle j}  : d ( i ∪ j ) , k = d i , k + d j , k 2 {\displaystyle d_{(i\cup j),k}={\frac {d_{i,k}+d_{j,k}}{2}}} The WPGMA algorithm produces rooted dendrograms and requires

2368-485: Is taken to be the average of all distances d ( x , y ) {\displaystyle d(x,y)} between pairs of objects x {\displaystyle x} in A {\displaystyle {\mathcal {A}}} and y {\displaystyle y} in B {\displaystyle {\mathcal {B}}} , that is, the mean distance between elements of each cluster: In other words, at each clustering step,

2432-779: Is the smallest value of D 3 {\displaystyle D_{3}} , so we join elements c {\displaystyle c} and d {\displaystyle d} . Let w {\displaystyle w} denote the node to which c {\displaystyle c} and d {\displaystyle d} are now connected. The branches joining c {\displaystyle c} and d {\displaystyle d} to w {\displaystyle w} then have lengths δ ( c , w ) = δ ( d , w ) = 28 / 2 = 14 {\displaystyle \delta (c,w)=\delta (d,w)=28/2=14} ( see

2496-779: Is the smallest value of D 3 {\displaystyle D_{3}} , so we join elements c {\displaystyle c} and d {\displaystyle d} . Let w {\displaystyle w} denote the node to which c {\displaystyle c} and d {\displaystyle d} are now connected. The branches joining c {\displaystyle c} and d {\displaystyle d} to w {\displaystyle w} then have lengths δ ( c , w ) = δ ( d , w ) = 28 / 2 = 14 {\displaystyle \delta (c,w)=\delta (d,w)=28/2=14} ( see

2560-442: Is therefore rooted by r {\displaystyle r} , its deepest node. Alternative linkage schemes include single linkage clustering , complete linkage clustering , and UPGMA average linkage clustering . Implementing a different linkage is simply a matter of using a different formula to calculate inter-cluster distances during the distance matrix update steps of the above algorithm. Complete linkage clustering avoids

2624-442: Is therefore rooted by r {\displaystyle r} , its deepest node. Alternative linkage schemes include single linkage clustering , complete linkage clustering , and WPGMA average linkage clustering . Implementing a different linkage is simply a matter of using a different formula to calculate inter-cluster distances during the distance matrix update steps of the above algorithm. Complete linkage clustering avoids

WPGMA - Misplaced Pages Continue

2688-498: The 5S ribosomal RNA sequence alignment of five bacteria: Bacillus subtilis ( a {\displaystyle a} ), Bacillus stearothermophilus ( b {\displaystyle b} ), Lactobacillus viridescens ( c {\displaystyle c} ), Acholeplasma modicum ( d {\displaystyle d} ), and Micrococcus luteus ( e {\displaystyle e} ). Let us assume that we have five elements (

2752-661: The algorithm to construct the UPGMA tree has O ( n 3 ) {\displaystyle O(n^{3})} time complexity, and using a heap for each cluster to keep its distances from other cluster reduces its time to O ( n 2 log ⁡ n ) {\displaystyle O(n^{2}\log n)} . Fionn Murtagh presented an O ( n 2 ) {\displaystyle O(n^{2})} time and space algorithm. Bacillus stearothermophilus Geobacillus stearothermophilus (previously Bacillus stearothermophilus )

2816-825: The calculation of this new distance accounts for the larger size of the ( a , b ) {\displaystyle (a,b)} cluster (two elements) with respect to e {\displaystyle e} (one element). Similarly: D 3 ( ( ( a , b ) , e ) , d ) = ( D 2 ( ( a , b ) , d ) × 2 + D 2 ( e , d ) × 1 ) / ( 2 + 1 ) = ( 32.5 × 2 + 43 × 1 ) / 3 = 36 {\displaystyle D_{3}(((a,b),e),d)=(D_{2}((a,b),d)\times 2+D_{2}(e,d)\times 1)/(2+1)=(32.5\times 2+43\times 1)/3=36} Proportional averaging therefore gives equal weight to

2880-448: The clustering of a {\displaystyle a} with b {\displaystyle b} . Bold values in D 2 {\displaystyle D_{2}} correspond to the new distances, calculated by averaging distances between each element of the first cluster ( a , b ) {\displaystyle (a,b)} and each of the remaining elements: D 2 ( (

2944-448: The clustering of a {\displaystyle a} with b {\displaystyle b} . Bold values in D 2 {\displaystyle D_{2}} correspond to the new distances, calculated by averaging distances between each element of the first cluster ( a , b ) {\displaystyle (a,b)} and each of the remaining elements: D 2 ( (

3008-464: The final dendrogram ) There is a single entry to update, keeping in mind that the two elements c {\displaystyle c} and d {\displaystyle d} each have a contribution of 1 {\displaystyle 1} in the average computation : D 4 ( ( c , d ) , ( ( a , b ) , e ) ) = ( D 3 ( c , ( (

3072-674: The final dendrogram ) There is a single entry to update: D 4 ( ( c , d ) , ( ( a , b ) , e ) ) = ( D 3 ( c , ( ( a , b ) , e ) ) + D 3 ( d , ( ( a , b ) , e ) ) ) / 2 = ( 32.25 + 37.75 ) / 2 = 35 {\displaystyle D_{4}((c,d),((a,b),e))=(D_{3}(c,((a,b),e))+D_{3}(d,((a,b),e)))/2=(32.25+37.75)/2=35} The final D 4 {\displaystyle D_{4}} matrix is: So we join clusters ( (

3136-502: The final dendrogram ) We then proceed to update D 2 {\displaystyle D_{2}} into a new distance matrix D 3 {\displaystyle D_{3}} (see below), reduced in size by one row and one column because of the clustering of ( a , b ) {\displaystyle (a,b)} with e {\displaystyle e} . Bold values in D 3 {\displaystyle D_{3}} correspond to

3200-467: The final dendrogram ) We then proceed to update the D 2 {\displaystyle D_{2}} matrix into a new distance matrix D 3 {\displaystyle D_{3}} (see below), reduced in size by one row and one column because of the clustering of ( a , b ) {\displaystyle (a,b)} with e {\displaystyle e}  : D 3 ( ( (

3264-424: The first cluster. We now reiterate the three previous steps, starting from the new distance matrix D 2 {\displaystyle D_{2}} Here, D 2 ( ( a , b ) , e ) = 22 {\displaystyle D_{2}((a,b),e)=22} is the smallest value of D 2 {\displaystyle D_{2}} , so we join cluster (

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3328-496: The initial distances of matrix D 1 {\displaystyle D_{1}} . This is the reason why the method is unweighted , not with respect to the mathematical procedure but with respect to the initial distances. We again reiterate the three previous steps, starting from the updated distance matrix D 3 {\displaystyle D_{3}} . Here, D 3 ( c , d ) = 28 {\displaystyle D_{3}(c,d)=28}

3392-494: The initial distances of matrix D 1 {\displaystyle D_{1}} . This is the reason why the method is weighted , not with respect to the mathematical procedure but with respect to the initial distances. We again reiterate the three previous steps, starting from the updated distance matrix D 3 {\displaystyle D_{3}} . Here, D 3 ( c , d ) = 28 {\displaystyle D_{3}(c,d)=28}

3456-661: The larger size of the ( a , b ) {\displaystyle (a,b)} cluster (two elements) with respect to e {\displaystyle e} (one element). Similarly: D 3 ( ( ( a , b ) , e ) , d ) = ( D 2 ( ( a , b ) , d ) + D 2 ( e , d ) ) / 2 = ( 32.5 + 43 ) / 2 = 37.75 {\displaystyle D_{3}(((a,b),e),d)=(D_{2}((a,b),d)+D_{2}(e,d))/2=(32.5+43)/2=37.75} The averaging procedure therefore gives differential weight to

3520-515: The matrix update as they correspond to distances between elements not involved in the first cluster. We now reiterate the three previous steps, starting from the new distance matrix D 2 {\displaystyle D_{2}}  : Here, D 2 ( ( a , b ) , e ) = 22 {\displaystyle D_{2}((a,b),e)=22} is the smallest value of D 2 {\displaystyle D_{2}} , so we join cluster (

3584-417: The missing branch length: δ ( u , v ) = δ ( e , v ) − δ ( a , u ) = δ ( e , v ) − δ ( b , u ) = 11 − 8.5 = 2.5 {\displaystyle \delta (u,v)=\delta (e,v)-\delta (a,u)=\delta (e,v)-\delta (b,u)=11-8.5=2.5} ( see

3648-416: The missing branch length: δ ( u , v ) = δ ( e , v ) − δ ( a , u ) = δ ( e , v ) − δ ( b , u ) = 11 − 8.5 = 2.5 {\displaystyle \delta (u,v)=\delta (e,v)-\delta (a,u)=\delta (e,v)-\delta (b,u)=11-8.5=2.5} ( see

3712-612: The new distances, calculated by proportional averaging : D 3 ( ( ( a , b ) , e ) , c ) = ( D 2 ( ( a , b ) , c ) × 2 + D 2 ( e , c ) × 1 ) / ( 2 + 1 ) = ( 25.5 × 2 + 39 × 1 ) / 3 = 30 {\displaystyle D_{3}(((a,b),e),c)=(D_{2}((a,b),c)\times 2+D_{2}(e,c)\times 1)/(2+1)=(25.5\times 2+39\times 1)/3=30} Thanks to this proportional average,

3776-538: The organism on filter paper inside a vial. After sterilizing, the cap is closed, an ampule of growth medium inside of the vial is crushed and the whole vial is incubated . A color and/or turbidity change indicates the results of the sterilization process; no change indicates that the sterilization conditions were achieved, otherwise the growth of the spores indicates that the sterilization process has not been met. Fluorescent-tagged strains, known as rapid-read BIs, are becoming more common to verify sterilization, since

3840-746: The proportional averaging in UPGMA produces an unweighted result ( see the working example ). The UPGMA algorithm constructs a rooted tree ( dendrogram ) that reflects the structure present in a pairwise similarity matrix (or a dissimilarity matrix ). At each step, the nearest two clusters are combined into a higher-level cluster. The distance between any two clusters A {\displaystyle {\mathcal {A}}} and B {\displaystyle {\mathcal {B}}} , each of size ( i.e. , cardinality ) | A | {\displaystyle {|{\mathcal {A}}|}} and | B | {\displaystyle {|{\mathcal {B}}|}} ,

3904-661: The two remaining branch lengths: δ ( v , r ) = δ ( ( ( a , b ) , e ) , r ) − δ ( e , v ) = 16.5 − 11 = 5.5 {\displaystyle \delta (v,r)=\delta (((a,b),e),r)-\delta (e,v)=16.5-11=5.5} δ ( w , r ) = δ ( ( c , d ) , r ) − δ ( c , w ) = 16.5 − 14 = 2.5 {\displaystyle \delta (w,r)=\delta ((c,d),r)-\delta (c,w)=16.5-14=2.5} The dendrogram

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3968-661: The two remaining branch lengths: δ ( v , r ) = δ ( ( ( a , b ) , e ) , r ) − δ ( e , v ) = 17.5 − 11 = 6.5 {\displaystyle \delta (v,r)=\delta (((a,b),e),r)-\delta (e,v)=17.5-11=6.5} δ ( w , r ) = δ ( ( c , d ) , r ) − δ ( c , w ) = 17.5 − 14 = 3.5 {\displaystyle \delta (w,r)=\delta ((c,d),r)-\delta (c,w)=17.5-14=3.5} The dendrogram

4032-1080: The updated distance between the joined clusters A ∪ B {\displaystyle {\mathcal {A}}\cup {\mathcal {B}}} and a new cluster X {\displaystyle X} is given by the proportional averaging of the d A , X {\displaystyle d_{{\mathcal {A}},X}} and d B , X {\displaystyle d_{{\mathcal {B}},X}} distances: d ( A ∪ B ) , X = | A | ⋅ d A , X + | B | ⋅ d B , X | A | + | B | {\displaystyle d_{({\mathcal {A}}\cup {\mathcal {B}}),X}={\frac {|{\mathcal {A}}|\cdot d_{{\mathcal {A}},X}+|{\mathcal {B}}|\cdot d_{{\mathcal {B}},X}}{|{\mathcal {A}}|+|{\mathcal {B}}|}}} The UPGMA algorithm produces rooted dendrograms and requires

4096-509: The visible fluorescence appears in about one-tenth the time needed for pH-indicator color change and an inexpensive light sensor can detect the growing colonies. It was first described in 1920 as Bacillus stearothermophilus , but, together with Bacillus thermoglucosidasius , it was reclassified as a member of the genus Geobacillus in 2001. Recently, a DNA polymerase derived from these bacteria, Bst polymerase, has become important in molecular biology applications. Bst polymerase has

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