- Email: [email protected]
Linkage Map Integration
GENOMICS
36, 157–162 (1996) 0436
ARTICLE NO.
Linkage Map Integration A. COLLINS,*,1 J. TEAGUE,* B. J. KEATS,†
AND
N. E. MORTON*
*Human Genetics, University of Southampton, Princess Anne Hospital, Level G, Southampton SO16 5YA, United Kingdom; and †Department of Biometry and Genetics, Louisiana State University Centre, 1901 Perdido Street, New Orleans, Louisiana 70112 Received November 28, 1995; accepted May 1, 1996
The algorithms that drive the map/ program for locus-oriented linkage mapping are presented. They depend on the enhanced location database program ldb/ to specify an initial comprehensive map that includes all loci in the summary lod file. Subsequently the map may be edited or order constrained and is automatically improved by estimating the location of each locus conditional on the remainder, beginning with the most discrepant loci. Operating characteristics permit rapid and accurate construction of linkage maps with several hundred loci. The map/ program also performs nondisjunction mapping with tests of nonstandard recombination. We have released map/ on Internet as a source program in the C language together with the location database that now includes the LODSOURCE database. The anonymous ftp is cedar.genetics.soton.ac.uk and the World Wide Web address is http://cedar.genetics.soton.ac.uk/ public_html. q 1996 Academic Press, Inc.
stored as partial maps and LODs, which are integrated by objective algorithms into a summary map. The highest synthesis in the summary map is a composite location that combines genetic and physical data on the physical scale. After the era in which it was fashionable to call all maps ‘‘primary,’’ it has become common to term a map ‘‘integrated’’ without implying the sine qua non of objectivity that ldb/ adopts. Human gene maps now include hundreds of loci, far too many for integration at a chromosome workshop or otherwise unless supported by a location database. Reference to a location database allows improvement of dense but locally unreliable maps. Our approach exploits multiple pairwise mapping through the program map/. Here we consider only linkage maps (including nondisjunction maps), leaving physical mapping and integration of genetic and physical maps to papers in preparation. METHODS
INTRODUCTION
The Human Genome Initiative has failed to develop a location database that would permit integration of genetic and physical data. Therefore the current standard is a consensus map in which the position of each locus is supported by at least one member of a workshop, but the evidence on which this location is based may be obscure. To remedy this situation, several attempts to facilitate map integration have been made (Bishop, 1994). Most programs (e.g., ACeDB, SIGMA) are largely graphical, which may be useful for display of integrated maps but not for their construction. The location database program ldb and the extended, enhanced, version ldb/ (Morton et al., 1992), have been extensively used (Morton, 1991a; Collins et al., 1992; Morton et al., 1992; Lawrence et al., 1993; Wang et al., 1994), and their salient features are illustrated in recent papers (Collins et al., 1995; Forabosco et al., 1995) and on our World Wide Web page. Data are 1 To whom correspondence should be addressed. Telephone: 01703 796939. Fax: 01703 794346.
Generalities. Each locus has an attribute called location, a vector of genetic and physical assignments. The location vector includes a composite location constructed by combining physical and genetic maps (Morton et al., 1992). The process that creates this vector by objective algorithms is called map integration. Typically the vector includes sex-specific genetic locations in centimorgans (cM), a physical location in megabases (Mb), with an origin at the short arm telomere (ptr), the left and right bands of the cytogenetic interval (where ‘‘left’’ is the band closest to the p telomere), a regional assignment, and mouse homology (Morton et al., 1992). A summary map for a given chromosome contains the vectors of genetic and physical locations together with the composite map. Integration by ldb/ operates on four types of data (partial maps, LODs, radiation hybrid, and clonal data). Each data set is a file that includes the source, format, and tabular results. The summary lod file includes all available LODs. The top directory chromi for the ith chromosome contains the summary file map, a synoptic reference file ref, and ancillary files that contain parameters used by the program. Linkage map integration by map/ uses only the lod and map files. In the recent past we have been preoccupied with map connectivity, which can be provided by an STS content map, radiation hybrids, FISH, or projection of the cytogenetic assignment onto the genetic map. Now we assume that connectivity and density have been achieved, and the problem is to improve order and location and to insert new loci. For this purpose we have developed locus-oriented algorithms that estimate the position of a selected locus, holding the others fixed. A region or all loci may be selected. The procedure is
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TABLE 1 Locus-Oriented Linkage Mapping of Locus i For sex k and loci h, i (h x i), √ Å frequency of typing error. In error filtered map with location Sik Å Shk / dhiwhik ,
uh* ik Å observed recombination Å
H
g01(whik) for k Å 1, 2 (uh* i1 / uh* i2)/2 for k Å 3,
u*hik Å expected recombination Å 2 √ (1 0 √)(1 0 uh* ik) / uh* ik [1 0 2 √ (1 0 √)]. ln L Å Rhik[ln u*hik] / (Nhik 0 Rhik)[ln (1 0 u*hik)], where R is the equivalent number of recombinants and N is the equivalent number of meioses (Morton and Andrews, 1989). Ìu*hik/Ìuh* ik Å 1 0 4 √ (1 0 √), ÌSik/Ìwhik Å dhi Ì ln L/Ìu*hik Å Rhik/u*hik(1 0 u*hik) 0 Nhik/(1 0 u*hik) Ì ln L/ÌSik Å (Ì ln L/Ìu*hik)(Ìu*hik/Ìuh* ik)/(Ìwhik/Ìuh* ik)(ÌShik/Ìwhik) p(1 0 2p)(1 0 4p)
Ì whik/Ìuh* ik Å
3(1 0 2uh* ik)
/
16p(p 0 1)(2p 0 1) 3[1 / 4(uh* ik) ] 2
/
2p(1 0 p)(8p / 2)
/ (1 0 p)(1 0 2p)(1 0 4p)
3[1 0 4(uh* ik)2]
(Rao et al., 1977). This is expressed in morgans and must be multiplied by 100 to give cM. For k Å 1, 2,
S D G S D S DS D G S D
uhik Å [Ì ln Lk/Ìu*hik / 0.5 Ì ln L3/Ìu*hi3][1 0 4 √ (1 0 √)]/dhi
Ihi11 Å
Ihi12 Å
Ihi22 Å
F F F
Nhi1
0.25Nhi3
/
u*hi1(1 0 u*hi1) 0.25 Nhi3
G
[1 0 4 √ (1 0 √)]2/
Nhi2
0.25 Nhi3
/
u*hi2 (1 0 u*hi2)
Ìwhi1
Ìwhi2
Ìuh* i1
Ìuh* i2
Ìuh* ik
Ìwhi1
[1 0 4 √ (1 0 √)]2/
2
Ìuh* i1
u*hi3(1 0 u*hi3)
u*hi3(1 0 u*hi3)
Ìwhik
[1 0 4 √ (1 0 √)]2/
u*hi3(1 0 u*hi3)
Ìwhi2
2
Ìuh* i2
uik Å ∑ uhik , Kikk* Å ∑ Iikk* , D Å Ki11Ki22 0 (Ki12)2.
Si1 r
Iteration:
Si2 r
Convergence x2 Å
Si1
if Ki11 Å 0
Si1 / (ui1Ki22 0 ui2 Ki12)/D
if D ú 0.1 Ki11Ki22
Si1 / ui1/Ki11
else.
Si2
if Ki22 Å 0
Si2 / (ui2 Ki11 0 ui1Ki12)/D
if D ú 0.1 Ki11Ki22
Si2 / ui2 /Ki22
else.
(ui1 / ui2 )2 /(Ki11 / Ki22 / 2Ki12 )
if Ki22 , Ki11 x 0, D õ 0.1 Ki11Ki22
2 ui1 /Ki11
if Ki22 Å 0, Ki11 ú 0
2 ui2 /Ki22
if Ki11 Å 0, Ki22 ú 0
2 2 Ki22 / ui2 Ki11 0 2ui1 ui2 Ki12 )/ D (ui1
else.
Heterogeneity x Å (h [(uhi1 / uhi2) /(Ihi11 / Ihi22 / 2Ii12)] 2
2
df Å (number of values of h) 0 1 q
Standard errors:
SE(Si1) Å Ki22 / D
if D ú 0.1 Ki11Ki22
q
1/Ki11
else
q
SE(Si2) Å Ki11/D
if D ú 0.1 Ki11Ki22
q
1/Ki22
else.
As soon as a cycle is completed without change of order, the logic switches to Table 2.
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TABLE 2
TABLE 3
Locus-Oriented Mapping with Order Fixed
Error Filtration and Interference
Let dhi Å
H
1
if the ith interval is included in segment h
0
else.
Given a map for fixed error frequency √ and mapping parameter p,
uh* k Å recombination over segment h for sex k
u(√)hik Å
w*ik Å length of ith interval in cM for sex k Ìu*hik Å Ì√
w*hk Å length of segment h for sex k.
S
H
Ìu*hik
F
Ì ln Lk
/
Ìu*hk
GS DYS D
0.5Ì ln L3
Ìu*hk
Ìw*hk
Ìu*h3
Ìuh* k
Ìuh* k
Ìu*hik Ì√
2(1 0 2√)(1 0 2uh* ik) for u* õ 0.499 0 else
The ML score for w*ik in a segment h for sex k Å 1, 2 is
uhik Å dhi
DS D
Ì ln Lk
3
u(√) Å ∑ ∑ u(√)hik hõi kÅ1
,
3
K(√) Å ∑ ∑ hõi kÅ1
F
Nhik
u*hik(1 0 u*hik)
GS D
Ìu*hik 2 . Ì√
where df Å (number of values of dhi Å 1) 0 2 Since w is a function of p and u it follows that (Ìw/Ìp)(Ìp/Ìu)(Ìu/ Ìw) Å 01. Therefore Ìu/Ìp Å 0(Ìw/Ìp)(Ìu/Ìw) and so
Sjk Å ∑ w*ik . i, j
Iteration on w*ik is the same as for Sik in Table 1, with constraint wik § 0. See Table 1 for U and K. After convergence a modified bootstrap is performed with this algorithm, changing the order of pairs of unconstrained loci and retaining those that reduce x2 by 1 or more; this is continued until a complete cycle gives no further change (Morton and Andrews, 1989).
0 u(p)hik Å
S
DS DS DYS D
Ì ln Lk
Ìu*hik
Ìu*hik
Ìuh* ik
Ìwhik Ìp
Ìwhik Ìuh* ik
for u* õ 0.499
0 else 3
u(p) Å ∑ ∑ u(p)hik hõi kÅ1 3
extremely fast and gives standard errors conditional on a fixed map. In addition to tabular output, this defines the location component of a locus catalog that would subsume relevant parts of McKusick (1994) and excel standards achieved for experimental organisms (Lindsley and Zimm, 1992; O’Brien, 1994). For locus-oriented integration we have avoided algorithms that require data other than location or distinguish framework from locally unordered loci. Each user is therefore free to choose loci on the basis of location, standard error, heterozygosity, and other characteristics that are most useful for a given application of the map. Linkage mapping. Partial lod files Lj for datasets j Å 1, . . . , have been described (Morton et al., 1992). They now include the LODSOURCE database (Keats et al., 1991), converted to a partial lod file in which sexes appear on the same record with references at the head of the file. Partial lod files are pooled into the summary lod file by ldb/. Multiple pairwise mapping is performed by map/, a new locus-oriented program. The program shares a small number of features with the earlier map program including the conversion of LODs to equivalent numbers of recombinants (R) and meioses (N) (Morton and Andrews, 1989; Morton and Collins, 1990). For pairs of loci with multiple entries in the lod file LODs are combined, with a heterogeneity test that assumes binomial sampling with sexes pooled and reports values of x2 with P õ 0.001 and contributions to x2 in excess of 10. The program requires a trial genetic map (as a job file) that contains locations for all or a subset of loci with LODs, which we take from the summary map generated by ldb/. To begin iteration for loci with LODs but not in the summary genetic map, a locus with a location in the composite map is interpolated into the sexspecific genetic map (Morton et al., 1992) but a locus without a composite location is given the location of the marker with highest LOD. Of course there is only one genetic location (fcm) for the X chromosome. The set of loci (h Å 1, 2, . . .) with pooled LODs to a selected locus i have locations Shk in the summary map, where k Å 1 for males and 2 for females. For unspecified sex, Sh3 Å (Sh1 / Sh2)/2. The location
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K(p) Å ∑ ∑ hõi kÅ1
F
Nhik
G
[1 0 4 √ (1 0 √)]2
u*hik(1 0 u*hik)
S D YS D Ìwhik Ìp
2
Ìwhik
2
Ìuh* ik
Ìwhik Å {a[ln(1 0 2uh* ik)] / b[tan01(2uh* ik)] Ìp / c[tanh01(2uh* ik) / duh* ik}/6 § 0, where a Å 12p 0 24p2 0 1 b Å 96p2 0 96p / 16 c Å 24p 0 48p2 / 4 d Å 168p 0 144p2 0 42 (Rao et al., 1977). 3
K(p√) Å ∑ ∑ hõi kÅ1
F
Nhik
GS DS DS DYS D
u*hik(1 0 u*hik)
Ìu*hik
Ìuh* ik
Ìwhik Ìp
Ìu*hik Ì√
Ìwhik Ìuh* ik
.
Single parameter iteration: p r p / u(p)/K(p) √ r √ / u(√)/K(√) convergence x12 Å u2/K 2 0 parameter estimation: D Å K(p)K(√) 0 [K(p√)]2 p r p / [u(p)K(√) 0 u(√)K(p√)]/D √ r √ / [u(√)K(p) 0 u(p)K(p√)]/D Convergence x2 Å [u(p)2K(√) / u(√)2K(p) 0 2u(p)u(√)K(p√)]/D.
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TABLE 4 Nondisjunction Mapping Following Morton et al. (1985), assume that true tetratype frequency y is related to recombination and map distance as y*hik Å 3uh* ik 0 whik but error frequency √ gives y*hik Å 2 √ (1 0 √)(1 0 y*hik) / y*hik[1 0 2 √ (1 0 √)], where k denotes a subset of the data (pat mei I, mat mei II, etc). The likelihood is a function of y, given by Table 4 of Shahar and Morton (1986) for eight mating types. Then
uhik Å
S
DS DS DYS D
Ì ln L Ì ln L Å ÌSik Ìy*hik
Ìy*hik
Ìy*hik Ìwhik
Ìy*hik
ÌSik , Ìwhik
where 3 0 Ìwhik/Ìuh* ik) Ìy*hik Å Ìwhik (Ìwhik/Ìuh* ik) Ìy* hik
Å 1 0 4 √ (1 0 √).
Ìy*hik Assuming √ and p are the same as for the standard map, an interesting hypothesis is that Sik Å MS*ik , where S*ik is the standard map for that sex. This hypothesis is most easily tested by interpolation on ˆ . The null hypothesis M Å 1 may be tested against the ln L to give M alternative hypothesis M and the general hypothesis in Sik , for which Sik r Sik / uik/Kik , where uik Å ∑ uhik h
S
Kik Å ∑ 0 h
DS D S D YS D
Ì2 ln L 2 hik
Ìy*
2
Ìy*hik
Ìy*hik
Ìy*hik Ìwhik
2
ÌS*ik 2 . Ìwhik
Likelihoods for pairs of loci each scored as R, N, D, or X (Morton et al., 1990) are defined as:
Locus pair
Likelihood
Observed number of pairs
N, N R, N or N, R or D, N or N, D R, R or R, D or D, R N, X or X, N R, X or X, R
2 0 y*hik y*hik 1 0 y*hik 4 0 y*hik 1 / y*hik
ahik bhik chik dhik ehik
Ì ln L
Å0
0
2 Ìy* hik
Å
ahik (2 0 y*hik)2 /
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2 0 y*hik
Ìy*hik Ì2ln L
ahik
/
dhik
/
bhik (y*hik)2 /
chik
0
y*hik
(4 0 y*hik)2
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dhik
0
4 0 y*hik
/
ehik 1 / y*hik
chik (1 0 y*hik)2
ehik (1 / y*hik)2
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Sik of i is estimated by Shk / dhi Whik , where dhi Å 1, 01 according to whether h is to the left or at the same location (Shk Å Sik), or to the right of i, and whik is the distance between h and i estimated from uhik . Where necessary, male and female locations are reestimated by interpolation using the sex-averaged location to ensure that they have the same flanking markers (Morton et al., 1992). An iterative estimate of the location of i maximizes the logarithmic likelihood ln L and allows movement outside the trial flanking markers (Table 1). At convergence the denominator of this expression gives a standard error (SEik) conditional on the rest of the map. The standard error on location indicates support for order. A subset of the loci may be indicated in the job file as order constrained such that all loci in the constrained set will retain their starting order. This allows reordering of a region of interest within a fixed map. Once order has stabilized, iteration continues with fixed order (Table 2). This process reestimates interlocus distances once the optimal order has been reached. After convergence under the order constraints in the job file, the analysis is repeated without constraints. The significance of order violations by the linkage map is tested by taking the difference in 2 ln L as a x2 with degrees of freedom equal to the number of constrained loci. Loci not represented in the summary lod file may later be incorporated into the summary map from partial maps by ldb/. Interference and error filtration. In multiple pairwise mapping interference is well described by the Rao mapping function with parameter p (Rao et al., 1977). There is no practical extension to multipoint mapping, in which order is determined on the assumption of no interference and distances under that hypothesis are subsequently adjusted to arbitrary interference. Aspects of the Rao function have recently been discussed (Morton, 1995). In map/ we impose the constraints w § u, Ìw/Ìp ú 0 (Weeks, 1994). Error filtration makes allowance for an error frequency √, assumed constant among loci (Shields et al., 1992). Like the mapping function, this is only an approximation, for which multipoint methods offer no parallel. Obviously correction of errors detected through blind replication is much preferable, but virtually never practiced. Selective retesting of outcomes with low probability is preferable to no correction but omission of discrepant observations is virtually useless as error filtration. Overestimating the mapping parameter (underestimating interference) is a common error that systematically inflates map length, as does neglect of error filtration. Disagreement with chiasma counts in males is largely due to these errors (Hulten, 1974; Morton, 1991b). Interference and error filtration interact, so that the false assumption of √ Å 0 inflates p. Whereas earlier studies that took √ Å 0 gave remarkably good agreement with p Å 0.35 (Morton et al., 1985), recent evidence indicates stronger interference on small chromosome arms and an error frequency in excess of 1% in uncontrolled data (Collins et al., 1995). In the presence of error multiple pairwise analysis gives more reliable maps than multipoint analysis (Buetow, 1991). The theory for estimating p and/or √ with order fixed is given in Table 3. Nondisjunction mapping. The origin of trisomy is exceedingly complex, with different mechanisms being important for different chromosomes. Nondisjunction is more frequent in females than in males and is associated with alterations in the level of recombination. This is expressed in terms of the tetratype frequency y, the probability in a single tetrad of obtaining all four possible chromatids generated by two heterozygous loci (Shahar and Morton, 1986; Morton et al., 1990). If the number of chromatid exchanges does not exceed 2, the tetratype frequency is related to recombination u and map distance w as y Å 3u 0 w. In map/ this theory is applied to data scored according to Table 1 of Morton et al. (1990), a typical example of which is given in their Table 5. For a selected subset (mat mei I, pat mei II, etc) the quantity Ì ln L/Ìy is evaluated in the eight mating types considered by Shahar and Morton (1986) with error filtration and transformed to maximum likelihood scores for map locations. Provision is made not only to fit the general hypothesis, estimating all locations, but also the subhy-
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TABLE 5 Comparison of map and map/ for Chromosome 9 x2
Final map length Male (cM)
Female (cM)
Start
Final
Timea (cpu min)
1. Correct starting order map map/
127.5 123.0 { 8.5
181.4 173.0 { 21.9
908.2 916.9
908.2 910.2
14.6 21.8
2. Poor starting order map map/
127.8 123.0 { 8.5
181.2 173.0 { 21.9
1184.8 1516.8
910.5 910.2
49.8 22.9
3. Very poor starting order map map/
126.7 123.0 { 8.5
303.6 173.0 { 21.9
6821.6 8227.0
4169.7 910.2
573.8 23.9
Program
a
On SUN SparcStation 10.
pothesis that specifies proportionality M between the nondisjunction map and the standard map for that parental sex (Table 4). As map length increases the value of y approaches 32 with damped oscillations and so a truncation parameter T may be specified to ignore pairs of loci with distance greater than T cM on the standard map for that sex, with default 50. T may also be used in general linkage mapping, where the default is no truncation. Comparison of methods. To provide an illustration of how this new approach compares to our previous map program we considered a 37-locus framework map of chromosome 9 assuming p Å 0.07 and √ Å 0.01 (Collins et al., 1995) using, primarily, CEPH 7 data (Dausset et al., 1990). Trial orders (correct, poor, and very poor) were submitted to the map bootstrap (Morton and Andrews, 1989). This performs pairwise flipping of loci and accepts improvements in order corresponding to a LOD of 1.0 or greater, the process being reinitialized after each change. The same orders were submitted to map/, which used the locus-oriented approach (as described in this paper) with a simplified bootstrap (one pass with improvements greater than a LOD of 1.0 accepted and the process continued to the end of the map). For the chromosome 9 framework loci both programs give similar maps (x2 Å 908.2 vs 910.2) and favor the same order (Table 5). The small discrepancy in x2 is accounted for by constraints w ú u and Ìw/Ìp ú 0 in map/ not imposed on map. For the correct starting order map/ is actually somewhat slower as it performs the locusoriented logic as well as the bootstrap. For a poor starting order map/ takes only 23 min (1 min more than the results for the correct starting order), but map takes more than twice as long to convergence (50 min). For a very poor starting order map fails to recover the correct locus order under these conditions and requires 9.5 h to complete the pairwise bootstrap but remains well away from the optimal solution. Map/, however, requires only 24 min to recover the optimal order. Map/ has been used to construct provisional, high-resolution maps of most human chromosomes. The data used were sex-specific LODs generated from the CEPH 7.1 database using a local version of the CRIMAP program (Lander and Green, 1987), combined with the LODSOURCE database (Keats et al., 1991). A comparison of all of these maps with the recent Ge´ne´thon linkage map (Dib et al., 1996) reveals only small numbers of local order discrepancies for markers shared between the two maps.
DISCUSSION
We have found that the above theory rapidly corrects an erroneous map and in a few iterations obtains an optimal order that would require much more human
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and computer time by other methods, if it could be reached at all. The multiple maxima characteristic of location scores do not arise in our pairwise method. Once order is stable on two successive cycles, map/ makes further improvement of sex specific locations with order fixed. In this way analyses that formerly took weeks now take hours. The results agree closely with much more time-consuming analyses by older methods, the advantage increasingly rapidly with the number of loci in the map. No comparison with programs like LINKAGE, MAPMAKER, or CRIMAP is possible, since they do not attempt to improve locally unreliable maps. The program MultiMap (Matise et al., 1994) is designed for automated multipoint map construction using CRIMAP for computation of likelihoods. This approach cannot, however, incorporate data available as pairwise LODs (for example, disease genes), does not include error filtration and interference in map construction, and does not have an interface with a location database. With this experience we are extending map integration in three directions. (1) Analysis of radiation hybrids and other sources of locus content maps is included in the map/ program, taking the data from ldb/, where it is retained for each chromosome. Raw hybrid data are stored as files prefixed ‘‘r.’’ Variable retention, polysomy, and error filtration are allowed for. (2) Clones available for contig construction are stored as ‘‘c’’ files. Physical map integration combines data from these clones and partial maps to construct a vector of physical locations. (3) The map//ldb/ suite incorporates all types of location data and analysis currently available and integrates them into a summary map that includes conditional standard errors on locations. These objective procedures conserve the evidence that validates the summary map but exhaustive and
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error-free entry of evidence on all chromosomes requires the efforts of the genome community. REFERENCES Bishop, M. (Ed.) (1994). ‘‘Guide to Human Genome Computing,’’ Academic Press, London. Buetow, K. H. (1991). Influence of aberrant observations on highresolution linkage analysis outcomes. Am. J. Hum. Genet. 49: 985– 994. Collins, A., Forabosco, P., Lawrence, S., and Morton, N. E. (1995). An integrated map of chromosome 9. Ann. Hum. Genet. 59: 393– 402. Collins, A., Keats, B. J., Dracopoli, N., Shields, D. C., and Morton, N. E. (1992). Integration of gene maps: Chromosome 1. Proc. Natl. Acad. Sci. USA 89: 4598–4602. Dausset, J., Cann, H., Cohen, D., Lathrop, M., Labouel, J. M., and White, R. (1990). Centre d’Etude du Polymorphisme Humain (CEPH) Collaborative mapping of the human genome. Genomics 6: 575–577. Dib, C., Faure, S., Fizames, C., Samson, D., Drouot, N., Vignal, A., Millasseau, P., Marc, S., Hazan, J., Seboun, E., Lathrop, M., Gyapay, G., Morissette, J., and Weissenbach, J. (1996). The Ge´ne´thon human genetic linkage map. Nature 380: 152–154. Forabosco, P., Collins, A., and Morton, N. E. (1995). Integration of gene maps: Updating chromosome 1. Ann. Hum. Genet. 59: 291– 305. Hulten, M. (1974). Chiasma distribution at diakinesis in the normal human male. Hereditas 76: 55–78. Keats, B. J. B., Sherman, S. L., Morton, N. E., Robson, E. B., Beutow, K. H., Cartwright, P. E., Chakravarti, A., Francke, U., Green, P. P., and Ott, J. (1991). Guidelines for human linkage maps. An International System for Human Linkage Maps (ISLM, 1990). Ann. Hum. Genet. 55: 106. Lander, E. S., and Green, P. (1987). Construction of multilocus genetic linkage maps in humans. Proc. Natl. Acad. Sci. USA 84: 2363–2367. Lawrence, S., Collins, A., Keats, B. J., Hulten, M., and Morton, N. E. (1993). Integration of gene maps: Chromosome 21. Proc. Natl. Acad. Sci. USA 90: 7210–7214.
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Lindsley, D. L., and Zimm, G. G. (1992). ‘‘The Genome of Drosophila melanogaster,’’ Academic Press, New York. Matise, T. C., Perlin, M., and Chakravarti, A. (1994). Automated construction of genetic linkage maps using an expert system (MultiMap): A human genome linkage map. Nature Genet. 6: 384– 390. McKusick, V. A. (1994). ‘‘Mendelian Inheritance in Man,’’ 11th ed., Johns Hopkins Press, Baltimore. Morton, N. E. (1991a). Gene maps and location databases. Ann. Hum. Genet. 55: 235–241. Morton, N. E. (1991b). Parameters of the human genome. Proc. Natl. Acad. Sci. USA 88: 7474–7476. Morton, N. E. (1995). Applicability of the Rao mapping function. Hum. Hered. 45: 178–180. Morton, N. E., and Andrews, V. (1989). MAP, an expert system for multiple pairwise linkage analysis. Ann. Hum. Genet. 53: 263– 269. Morton, N. E., and Collins, A. (1990). Standard maps of chromosome 10. Ann. Hum. Genet. 54: 235–251. Morton, N. E., Collins, A., Lawrence, S., and Shields, D. C. (1992). Algorithms for a location database. Ann. Hum. Genet. 56: 223– 232. Morton, N. E., Keats, B. J., Jacobs, P. A., Hassold, T., Pettay, D., Harvey, J., and Andrews, V. (1990). A centromere map of the X chromosome from trisomies of maternal origin. Ann. Hum. Genet. 54: 39–47. Morton, N. E., MacLean, C. J., and Lew, R. (1985). Tests of hypotheses on recombination frequencies. Genet. Res. 45: 279–286. O’Brien, S. J. (1994). ‘‘Genetic Maps. Locus Maps of Complex Genomes,’’ 6th ed., Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY. Rao, D. C., Morton, N. E., Lindsten, J., Hulten, M., and Yee, S. (1977). A mapping function for man. Hum. Hered. 27: 99–104. Shahar, S., and Morton, N. E. (1986). Origin of teratomas and twins. Hum. Genet. 74: 215–218. Shields, D. C., Collins, A., Buetow, K. H., and Morton, N. E. (1991). Error filtration, interference, and the human linkage map. Proc. Natl. Acad. Sci. USA 88: 6501–6505. Wang, L. H., Collins, A., Lawrence, S., Keats, B. J., and Morton, N. E. (1994). Integration of gene maps: Chromosome X. Genomics 22: 590–604. Weeks, D. E. (1994). Invalidity of the Rao map function for three loci. Hum. Hered. 44: 178–180.
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ARTICLE NO.
Linkage Map Integration A. COLLINS,*,1 J. TEAGUE,* B. J. KEATS,†
AND
N. E. MORTON*
*Human Genetics, University of Southampton, Princess Anne Hospital, Level G, Southampton SO16 5YA, United Kingdom; and †Department of Biometry and Genetics, Louisiana State University Centre, 1901 Perdido Street, New Orleans, Louisiana 70112 Received November 28, 1995; accepted May 1, 1996
The algorithms that drive the map/ program for locus-oriented linkage mapping are presented. They depend on the enhanced location database program ldb/ to specify an initial comprehensive map that includes all loci in the summary lod file. Subsequently the map may be edited or order constrained and is automatically improved by estimating the location of each locus conditional on the remainder, beginning with the most discrepant loci. Operating characteristics permit rapid and accurate construction of linkage maps with several hundred loci. The map/ program also performs nondisjunction mapping with tests of nonstandard recombination. We have released map/ on Internet as a source program in the C language together with the location database that now includes the LODSOURCE database. The anonymous ftp is cedar.genetics.soton.ac.uk and the World Wide Web address is http://cedar.genetics.soton.ac.uk/ public_html. q 1996 Academic Press, Inc.
stored as partial maps and LODs, which are integrated by objective algorithms into a summary map. The highest synthesis in the summary map is a composite location that combines genetic and physical data on the physical scale. After the era in which it was fashionable to call all maps ‘‘primary,’’ it has become common to term a map ‘‘integrated’’ without implying the sine qua non of objectivity that ldb/ adopts. Human gene maps now include hundreds of loci, far too many for integration at a chromosome workshop or otherwise unless supported by a location database. Reference to a location database allows improvement of dense but locally unreliable maps. Our approach exploits multiple pairwise mapping through the program map/. Here we consider only linkage maps (including nondisjunction maps), leaving physical mapping and integration of genetic and physical maps to papers in preparation. METHODS
INTRODUCTION
The Human Genome Initiative has failed to develop a location database that would permit integration of genetic and physical data. Therefore the current standard is a consensus map in which the position of each locus is supported by at least one member of a workshop, but the evidence on which this location is based may be obscure. To remedy this situation, several attempts to facilitate map integration have been made (Bishop, 1994). Most programs (e.g., ACeDB, SIGMA) are largely graphical, which may be useful for display of integrated maps but not for their construction. The location database program ldb and the extended, enhanced, version ldb/ (Morton et al., 1992), have been extensively used (Morton, 1991a; Collins et al., 1992; Morton et al., 1992; Lawrence et al., 1993; Wang et al., 1994), and their salient features are illustrated in recent papers (Collins et al., 1995; Forabosco et al., 1995) and on our World Wide Web page. Data are 1 To whom correspondence should be addressed. Telephone: 01703 796939. Fax: 01703 794346.
Generalities. Each locus has an attribute called location, a vector of genetic and physical assignments. The location vector includes a composite location constructed by combining physical and genetic maps (Morton et al., 1992). The process that creates this vector by objective algorithms is called map integration. Typically the vector includes sex-specific genetic locations in centimorgans (cM), a physical location in megabases (Mb), with an origin at the short arm telomere (ptr), the left and right bands of the cytogenetic interval (where ‘‘left’’ is the band closest to the p telomere), a regional assignment, and mouse homology (Morton et al., 1992). A summary map for a given chromosome contains the vectors of genetic and physical locations together with the composite map. Integration by ldb/ operates on four types of data (partial maps, LODs, radiation hybrid, and clonal data). Each data set is a file that includes the source, format, and tabular results. The summary lod file includes all available LODs. The top directory chromi for the ith chromosome contains the summary file map, a synoptic reference file ref, and ancillary files that contain parameters used by the program. Linkage map integration by map/ uses only the lod and map files. In the recent past we have been preoccupied with map connectivity, which can be provided by an STS content map, radiation hybrids, FISH, or projection of the cytogenetic assignment onto the genetic map. Now we assume that connectivity and density have been achieved, and the problem is to improve order and location and to insert new loci. For this purpose we have developed locus-oriented algorithms that estimate the position of a selected locus, holding the others fixed. A region or all loci may be selected. The procedure is
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TABLE 1 Locus-Oriented Linkage Mapping of Locus i For sex k and loci h, i (h x i), √ Å frequency of typing error. In error filtered map with location Sik Å Shk / dhiwhik ,
uh* ik Å observed recombination Å
H
g01(whik) for k Å 1, 2 (uh* i1 / uh* i2)/2 for k Å 3,
u*hik Å expected recombination Å 2 √ (1 0 √)(1 0 uh* ik) / uh* ik [1 0 2 √ (1 0 √)]. ln L Å Rhik[ln u*hik] / (Nhik 0 Rhik)[ln (1 0 u*hik)], where R is the equivalent number of recombinants and N is the equivalent number of meioses (Morton and Andrews, 1989). Ìu*hik/Ìuh* ik Å 1 0 4 √ (1 0 √), ÌSik/Ìwhik Å dhi Ì ln L/Ìu*hik Å Rhik/u*hik(1 0 u*hik) 0 Nhik/(1 0 u*hik) Ì ln L/ÌSik Å (Ì ln L/Ìu*hik)(Ìu*hik/Ìuh* ik)/(Ìwhik/Ìuh* ik)(ÌShik/Ìwhik) p(1 0 2p)(1 0 4p)
Ì whik/Ìuh* ik Å
3(1 0 2uh* ik)
/
16p(p 0 1)(2p 0 1) 3[1 / 4(uh* ik) ] 2
/
2p(1 0 p)(8p / 2)
/ (1 0 p)(1 0 2p)(1 0 4p)
3[1 0 4(uh* ik)2]
(Rao et al., 1977). This is expressed in morgans and must be multiplied by 100 to give cM. For k Å 1, 2,
S D G S D S DS D G S D
uhik Å [Ì ln Lk/Ìu*hik / 0.5 Ì ln L3/Ìu*hi3][1 0 4 √ (1 0 √)]/dhi
Ihi11 Å
Ihi12 Å
Ihi22 Å
F F F
Nhi1
0.25Nhi3
/
u*hi1(1 0 u*hi1) 0.25 Nhi3
G
[1 0 4 √ (1 0 √)]2/
Nhi2
0.25 Nhi3
/
u*hi2 (1 0 u*hi2)
Ìwhi1
Ìwhi2
Ìuh* i1
Ìuh* i2
Ìuh* ik
Ìwhi1
[1 0 4 √ (1 0 √)]2/
2
Ìuh* i1
u*hi3(1 0 u*hi3)
u*hi3(1 0 u*hi3)
Ìwhik
[1 0 4 √ (1 0 √)]2/
u*hi3(1 0 u*hi3)
Ìwhi2
2
Ìuh* i2
uik Å ∑ uhik , Kikk* Å ∑ Iikk* , D Å Ki11Ki22 0 (Ki12)2.
Si1 r
Iteration:
Si2 r
Convergence x2 Å
Si1
if Ki11 Å 0
Si1 / (ui1Ki22 0 ui2 Ki12)/D
if D ú 0.1 Ki11Ki22
Si1 / ui1/Ki11
else.
Si2
if Ki22 Å 0
Si2 / (ui2 Ki11 0 ui1Ki12)/D
if D ú 0.1 Ki11Ki22
Si2 / ui2 /Ki22
else.
(ui1 / ui2 )2 /(Ki11 / Ki22 / 2Ki12 )
if Ki22 , Ki11 x 0, D õ 0.1 Ki11Ki22
2 ui1 /Ki11
if Ki22 Å 0, Ki11 ú 0
2 ui2 /Ki22
if Ki11 Å 0, Ki22 ú 0
2 2 Ki22 / ui2 Ki11 0 2ui1 ui2 Ki12 )/ D (ui1
else.
Heterogeneity x Å (h [(uhi1 / uhi2) /(Ihi11 / Ihi22 / 2Ii12)] 2
2
df Å (number of values of h) 0 1 q
Standard errors:
SE(Si1) Å Ki22 / D
if D ú 0.1 Ki11Ki22
q
1/Ki11
else
q
SE(Si2) Å Ki11/D
if D ú 0.1 Ki11Ki22
q
1/Ki22
else.
As soon as a cycle is completed without change of order, the logic switches to Table 2.
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TABLE 2
TABLE 3
Locus-Oriented Mapping with Order Fixed
Error Filtration and Interference
Let dhi Å
H
1
if the ith interval is included in segment h
0
else.
Given a map for fixed error frequency √ and mapping parameter p,
uh* k Å recombination over segment h for sex k
u(√)hik Å
w*ik Å length of ith interval in cM for sex k Ìu*hik Å Ì√
w*hk Å length of segment h for sex k.
S
H
Ìu*hik
F
Ì ln Lk
/
Ìu*hk
GS DYS D
0.5Ì ln L3
Ìu*hk
Ìw*hk
Ìu*h3
Ìuh* k
Ìuh* k
Ìu*hik Ì√
2(1 0 2√)(1 0 2uh* ik) for u* õ 0.499 0 else
The ML score for w*ik in a segment h for sex k Å 1, 2 is
uhik Å dhi
DS D
Ì ln Lk
3
u(√) Å ∑ ∑ u(√)hik hõi kÅ1
,
3
K(√) Å ∑ ∑ hõi kÅ1
F
Nhik
u*hik(1 0 u*hik)
GS D
Ìu*hik 2 . Ì√
where df Å (number of values of dhi Å 1) 0 2 Since w is a function of p and u it follows that (Ìw/Ìp)(Ìp/Ìu)(Ìu/ Ìw) Å 01. Therefore Ìu/Ìp Å 0(Ìw/Ìp)(Ìu/Ìw) and so
Sjk Å ∑ w*ik . i, j
Iteration on w*ik is the same as for Sik in Table 1, with constraint wik § 0. See Table 1 for U and K. After convergence a modified bootstrap is performed with this algorithm, changing the order of pairs of unconstrained loci and retaining those that reduce x2 by 1 or more; this is continued until a complete cycle gives no further change (Morton and Andrews, 1989).
0 u(p)hik Å
S
DS DS DYS D
Ì ln Lk
Ìu*hik
Ìu*hik
Ìuh* ik
Ìwhik Ìp
Ìwhik Ìuh* ik
for u* õ 0.499
0 else 3
u(p) Å ∑ ∑ u(p)hik hõi kÅ1 3
extremely fast and gives standard errors conditional on a fixed map. In addition to tabular output, this defines the location component of a locus catalog that would subsume relevant parts of McKusick (1994) and excel standards achieved for experimental organisms (Lindsley and Zimm, 1992; O’Brien, 1994). For locus-oriented integration we have avoided algorithms that require data other than location or distinguish framework from locally unordered loci. Each user is therefore free to choose loci on the basis of location, standard error, heterozygosity, and other characteristics that are most useful for a given application of the map. Linkage mapping. Partial lod files Lj for datasets j Å 1, . . . , have been described (Morton et al., 1992). They now include the LODSOURCE database (Keats et al., 1991), converted to a partial lod file in which sexes appear on the same record with references at the head of the file. Partial lod files are pooled into the summary lod file by ldb/. Multiple pairwise mapping is performed by map/, a new locus-oriented program. The program shares a small number of features with the earlier map program including the conversion of LODs to equivalent numbers of recombinants (R) and meioses (N) (Morton and Andrews, 1989; Morton and Collins, 1990). For pairs of loci with multiple entries in the lod file LODs are combined, with a heterogeneity test that assumes binomial sampling with sexes pooled and reports values of x2 with P õ 0.001 and contributions to x2 in excess of 10. The program requires a trial genetic map (as a job file) that contains locations for all or a subset of loci with LODs, which we take from the summary map generated by ldb/. To begin iteration for loci with LODs but not in the summary genetic map, a locus with a location in the composite map is interpolated into the sexspecific genetic map (Morton et al., 1992) but a locus without a composite location is given the location of the marker with highest LOD. Of course there is only one genetic location (fcm) for the X chromosome. The set of loci (h Å 1, 2, . . .) with pooled LODs to a selected locus i have locations Shk in the summary map, where k Å 1 for males and 2 for females. For unspecified sex, Sh3 Å (Sh1 / Sh2)/2. The location
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K(p) Å ∑ ∑ hõi kÅ1
F
Nhik
G
[1 0 4 √ (1 0 √)]2
u*hik(1 0 u*hik)
S D YS D Ìwhik Ìp
2
Ìwhik
2
Ìuh* ik
Ìwhik Å {a[ln(1 0 2uh* ik)] / b[tan01(2uh* ik)] Ìp / c[tanh01(2uh* ik) / duh* ik}/6 § 0, where a Å 12p 0 24p2 0 1 b Å 96p2 0 96p / 16 c Å 24p 0 48p2 / 4 d Å 168p 0 144p2 0 42 (Rao et al., 1977). 3
K(p√) Å ∑ ∑ hõi kÅ1
F
Nhik
GS DS DS DYS D
u*hik(1 0 u*hik)
Ìu*hik
Ìuh* ik
Ìwhik Ìp
Ìu*hik Ì√
Ìwhik Ìuh* ik
.
Single parameter iteration: p r p / u(p)/K(p) √ r √ / u(√)/K(√) convergence x12 Å u2/K 2 0 parameter estimation: D Å K(p)K(√) 0 [K(p√)]2 p r p / [u(p)K(√) 0 u(√)K(p√)]/D √ r √ / [u(√)K(p) 0 u(p)K(p√)]/D Convergence x2 Å [u(p)2K(√) / u(√)2K(p) 0 2u(p)u(√)K(p√)]/D.
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TABLE 4 Nondisjunction Mapping Following Morton et al. (1985), assume that true tetratype frequency y is related to recombination and map distance as y*hik Å 3uh* ik 0 whik but error frequency √ gives y*hik Å 2 √ (1 0 √)(1 0 y*hik) / y*hik[1 0 2 √ (1 0 √)], where k denotes a subset of the data (pat mei I, mat mei II, etc). The likelihood is a function of y, given by Table 4 of Shahar and Morton (1986) for eight mating types. Then
uhik Å
S
DS DS DYS D
Ì ln L Ì ln L Å ÌSik Ìy*hik
Ìy*hik
Ìy*hik Ìwhik
Ìy*hik
ÌSik , Ìwhik
where 3 0 Ìwhik/Ìuh* ik) Ìy*hik Å Ìwhik (Ìwhik/Ìuh* ik) Ìy* hik
Å 1 0 4 √ (1 0 √).
Ìy*hik Assuming √ and p are the same as for the standard map, an interesting hypothesis is that Sik Å MS*ik , where S*ik is the standard map for that sex. This hypothesis is most easily tested by interpolation on ˆ . The null hypothesis M Å 1 may be tested against the ln L to give M alternative hypothesis M and the general hypothesis in Sik , for which Sik r Sik / uik/Kik , where uik Å ∑ uhik h
S
Kik Å ∑ 0 h
DS D S D YS D
Ì2 ln L 2 hik
Ìy*
2
Ìy*hik
Ìy*hik
Ìy*hik Ìwhik
2
ÌS*ik 2 . Ìwhik
Likelihoods for pairs of loci each scored as R, N, D, or X (Morton et al., 1990) are defined as:
Locus pair
Likelihood
Observed number of pairs
N, N R, N or N, R or D, N or N, D R, R or R, D or D, R N, X or X, N R, X or X, R
2 0 y*hik y*hik 1 0 y*hik 4 0 y*hik 1 / y*hik
ahik bhik chik dhik ehik
Ì ln L
Å0
0
2 Ìy* hik
Å
ahik (2 0 y*hik)2 /
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Ìy*hik Ì2ln L
ahik
/
dhik
/
bhik (y*hik)2 /
chik
0
y*hik
(4 0 y*hik)2
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Sik of i is estimated by Shk / dhi Whik , where dhi Å 1, 01 according to whether h is to the left or at the same location (Shk Å Sik), or to the right of i, and whik is the distance between h and i estimated from uhik . Where necessary, male and female locations are reestimated by interpolation using the sex-averaged location to ensure that they have the same flanking markers (Morton et al., 1992). An iterative estimate of the location of i maximizes the logarithmic likelihood ln L and allows movement outside the trial flanking markers (Table 1). At convergence the denominator of this expression gives a standard error (SEik) conditional on the rest of the map. The standard error on location indicates support for order. A subset of the loci may be indicated in the job file as order constrained such that all loci in the constrained set will retain their starting order. This allows reordering of a region of interest within a fixed map. Once order has stabilized, iteration continues with fixed order (Table 2). This process reestimates interlocus distances once the optimal order has been reached. After convergence under the order constraints in the job file, the analysis is repeated without constraints. The significance of order violations by the linkage map is tested by taking the difference in 2 ln L as a x2 with degrees of freedom equal to the number of constrained loci. Loci not represented in the summary lod file may later be incorporated into the summary map from partial maps by ldb/. Interference and error filtration. In multiple pairwise mapping interference is well described by the Rao mapping function with parameter p (Rao et al., 1977). There is no practical extension to multipoint mapping, in which order is determined on the assumption of no interference and distances under that hypothesis are subsequently adjusted to arbitrary interference. Aspects of the Rao function have recently been discussed (Morton, 1995). In map/ we impose the constraints w § u, Ìw/Ìp ú 0 (Weeks, 1994). Error filtration makes allowance for an error frequency √, assumed constant among loci (Shields et al., 1992). Like the mapping function, this is only an approximation, for which multipoint methods offer no parallel. Obviously correction of errors detected through blind replication is much preferable, but virtually never practiced. Selective retesting of outcomes with low probability is preferable to no correction but omission of discrepant observations is virtually useless as error filtration. Overestimating the mapping parameter (underestimating interference) is a common error that systematically inflates map length, as does neglect of error filtration. Disagreement with chiasma counts in males is largely due to these errors (Hulten, 1974; Morton, 1991b). Interference and error filtration interact, so that the false assumption of √ Å 0 inflates p. Whereas earlier studies that took √ Å 0 gave remarkably good agreement with p Å 0.35 (Morton et al., 1985), recent evidence indicates stronger interference on small chromosome arms and an error frequency in excess of 1% in uncontrolled data (Collins et al., 1995). In the presence of error multiple pairwise analysis gives more reliable maps than multipoint analysis (Buetow, 1991). The theory for estimating p and/or √ with order fixed is given in Table 3. Nondisjunction mapping. The origin of trisomy is exceedingly complex, with different mechanisms being important for different chromosomes. Nondisjunction is more frequent in females than in males and is associated with alterations in the level of recombination. This is expressed in terms of the tetratype frequency y, the probability in a single tetrad of obtaining all four possible chromatids generated by two heterozygous loci (Shahar and Morton, 1986; Morton et al., 1990). If the number of chromatid exchanges does not exceed 2, the tetratype frequency is related to recombination u and map distance w as y Å 3u 0 w. In map/ this theory is applied to data scored according to Table 1 of Morton et al. (1990), a typical example of which is given in their Table 5. For a selected subset (mat mei I, pat mei II, etc) the quantity Ì ln L/Ìy is evaluated in the eight mating types considered by Shahar and Morton (1986) with error filtration and transformed to maximum likelihood scores for map locations. Provision is made not only to fit the general hypothesis, estimating all locations, but also the subhy-
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TABLE 5 Comparison of map and map/ for Chromosome 9 x2
Final map length Male (cM)
Female (cM)
Start
Final
Timea (cpu min)
1. Correct starting order map map/
127.5 123.0 { 8.5
181.4 173.0 { 21.9
908.2 916.9
908.2 910.2
14.6 21.8
2. Poor starting order map map/
127.8 123.0 { 8.5
181.2 173.0 { 21.9
1184.8 1516.8
910.5 910.2
49.8 22.9
3. Very poor starting order map map/
126.7 123.0 { 8.5
303.6 173.0 { 21.9
6821.6 8227.0
4169.7 910.2
573.8 23.9
Program
a
On SUN SparcStation 10.
pothesis that specifies proportionality M between the nondisjunction map and the standard map for that parental sex (Table 4). As map length increases the value of y approaches 32 with damped oscillations and so a truncation parameter T may be specified to ignore pairs of loci with distance greater than T cM on the standard map for that sex, with default 50. T may also be used in general linkage mapping, where the default is no truncation. Comparison of methods. To provide an illustration of how this new approach compares to our previous map program we considered a 37-locus framework map of chromosome 9 assuming p Å 0.07 and √ Å 0.01 (Collins et al., 1995) using, primarily, CEPH 7 data (Dausset et al., 1990). Trial orders (correct, poor, and very poor) were submitted to the map bootstrap (Morton and Andrews, 1989). This performs pairwise flipping of loci and accepts improvements in order corresponding to a LOD of 1.0 or greater, the process being reinitialized after each change. The same orders were submitted to map/, which used the locus-oriented approach (as described in this paper) with a simplified bootstrap (one pass with improvements greater than a LOD of 1.0 accepted and the process continued to the end of the map). For the chromosome 9 framework loci both programs give similar maps (x2 Å 908.2 vs 910.2) and favor the same order (Table 5). The small discrepancy in x2 is accounted for by constraints w ú u and Ìw/Ìp ú 0 in map/ not imposed on map. For the correct starting order map/ is actually somewhat slower as it performs the locusoriented logic as well as the bootstrap. For a poor starting order map/ takes only 23 min (1 min more than the results for the correct starting order), but map takes more than twice as long to convergence (50 min). For a very poor starting order map fails to recover the correct locus order under these conditions and requires 9.5 h to complete the pairwise bootstrap but remains well away from the optimal solution. Map/, however, requires only 24 min to recover the optimal order. Map/ has been used to construct provisional, high-resolution maps of most human chromosomes. The data used were sex-specific LODs generated from the CEPH 7.1 database using a local version of the CRIMAP program (Lander and Green, 1987), combined with the LODSOURCE database (Keats et al., 1991). A comparison of all of these maps with the recent Ge´ne´thon linkage map (Dib et al., 1996) reveals only small numbers of local order discrepancies for markers shared between the two maps.
DISCUSSION
We have found that the above theory rapidly corrects an erroneous map and in a few iterations obtains an optimal order that would require much more human
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and computer time by other methods, if it could be reached at all. The multiple maxima characteristic of location scores do not arise in our pairwise method. Once order is stable on two successive cycles, map/ makes further improvement of sex specific locations with order fixed. In this way analyses that formerly took weeks now take hours. The results agree closely with much more time-consuming analyses by older methods, the advantage increasingly rapidly with the number of loci in the map. No comparison with programs like LINKAGE, MAPMAKER, or CRIMAP is possible, since they do not attempt to improve locally unreliable maps. The program MultiMap (Matise et al., 1994) is designed for automated multipoint map construction using CRIMAP for computation of likelihoods. This approach cannot, however, incorporate data available as pairwise LODs (for example, disease genes), does not include error filtration and interference in map construction, and does not have an interface with a location database. With this experience we are extending map integration in three directions. (1) Analysis of radiation hybrids and other sources of locus content maps is included in the map/ program, taking the data from ldb/, where it is retained for each chromosome. Raw hybrid data are stored as files prefixed ‘‘r.’’ Variable retention, polysomy, and error filtration are allowed for. (2) Clones available for contig construction are stored as ‘‘c’’ files. Physical map integration combines data from these clones and partial maps to construct a vector of physical locations. (3) The map//ldb/ suite incorporates all types of location data and analysis currently available and integrates them into a summary map that includes conditional standard errors on locations. These objective procedures conserve the evidence that validates the summary map but exhaustive and
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AP: Genomics