Genomics via optical mapping ordered restriction maps

A method of producing high-resolution, high-accuracy ordered restriction maps based on data created from the images of populations of individual DNA molecules (clones) digested by restriction enzymes. Detailed modeling and a statistical algorithm, along with an interactive algorithm based on dynamic programming and a heuristic method employing branch-and-bound procedures, are used to find the most likely true restriction map, based on experimental data.

BACKGROUND OF THE INVENTION
 This invention relates to the construction of ordered restriction maps and,
 more particularly, to a method capable of producing high-resolution,
 high-accuracy maps rapidly and in a scalable manner.
 Optical mapping is a single molecule methodology for the rapid production
 of ordered restriction maps from individual DNA molecules. Ordered
 restriction maps are constructed by using fluorescence microscopy to
 visualize restriction endonuclease cutting events on individual
 fluorochrome-stained DNA molecules. Restriction enzyme cleavage sites are
 visible as gaps that appear flanking the relaxed DNA fragments (pieces of
 molecules between two consecutive cleavages). Relative fluorescence
 intensity (measuring the amount of fluorochrome binding to the restriction
 fragment) or apparent length measurements (along a well-defined "backbone"
 spanning the restriction fragment) have proven to provide accurate
 size-estimates of the restriction fragment and have been used to construct
 the final restriction map.
 Such a restriction map created from one individual DNA molecule is limited
 in its accuracy by the resolution of the microscopy, the imaging system
 (CCD camera, quantization level, etc.), illumination and surface
 conditions. Furthermore, depending on the digestion rate and the noise
 inherent to the intensity distribution along the DNA molecule, with some
 probability one is likely to miss a small fraction of the restriction
 sites or introduce spurious sites. Additionally, investigators may
 sometimes (rather infrequently) lack the exact orientation information
 (whether the left-most restriction site is the first or the last). Thus,
 given two arbitrary single molecule restriction maps for the same DNA
 clone obtained this way, the maps are expected to be roughly the same in
 the following sense--if the maps are "aligned" by first choosing the
 orientation and then identifying the restrictions sites that differ by
 small amount, then most of the restrictions sites will appear roughly at
 the same place in both the maps.
 There are two approaches to further improve the accuracy and resolution of
 the maps: first, improve the chemical and optical processes to minimize
 the effect of each error source and second, use statistical approaches
 where the restriction maps of a large number of identical clones are
 combined to create a high-accuracy restriction map. These two approaches
 are not mutually exclusive and interesting trade-offs exist that can be
 exploited fruitfully.
 For instance, in the original method, fluorescently-labeled DNA molecules
 were elongated in a flow of molten agarose containing restriction
 endonucleases, generated between a cover-slip and a microscope slide, and
 the resulting cleavage events were recorded by fluorescence microscopy as
 time-lapse digitized images. The second generation optical mapping
 approach, which dispensed with agarose and time-lapsed imaging, involves
 fixing elongated DNA molecules onto positively-charged glass surfaces,
 thus improving sizing precision as well as throughput for a wide range of
 cloning vectors (cosmid, bacteriophage, and yeast or bacterial artificial
 chromosomes (YAC or BAC)). Further improvements have recently come from
 many sources: development of a simple and reliable procedure to mount
 large DNA molecules with good molecular extension and minimal breakage;
 optimization of the surface derivatization, maximizing the range of usable
 restriction enzymes and retention of small fragments; and development of
 an open surface digestion format, facilitating access to samples and
 laying the foundations for automated approaches to mapping large insert
 clones.
 Improvements have also come from powerful statistical tools that process a
 preliminary collection of single-molecule restriction maps, each one
 created from an image of a DNA molecule belonging to a pool of identical
 clones. Such restriction maps are almost identical with small variations
 resulting from sizing errors, partially digested restriction sites and
 "false" restriction sites. Such maps can be combined easily in most cases.
 However, the underlying statistical problem poses many fundamental
 challenges; for example, the presence of some uncertainty in the alignment
 of a molecule (both orientation and/or matching in the sites) in
 conjunction with either false cuts or sizing error is sufficient to make
 the problem infeasible (NP-complete). M. R. Garey and D. S. Johnson,
 Computer and Intractability: A Guide to the Theory of NP-Completeness (W.
 H. Freeman and Co., San Francisco 1979).
 In spite of the pessimistic results mentioned above, the problem admits
 efficient algorithms once the structure in the input is exploited. For
 instance, if the digestion rate is quite high, then by looking at the
 distribution of the cuts a good guess can be made about the number of cuts
 and then only the data set with large numbers of cuts can be combined to
 create the final map. J. Reed, Optical Mapping, Ph. D. Thesis, NYU, June
 1997 (Expected). Other approaches have used formulations in which one
 optimizes a cost function and provides heuristics (as the exact
 optimization problems are often infeasible). In one approach, the
 optimization problem corresponds to finding weighted cliques; and in
 another, the formulation corresponds to a 0-1 quadratic programming
 problem. S. Muthukrishnan and L. Parida. "Towards Constructing Physical
 Maps by Optical Mapping: An Effective Simple Combinatorial Approach, in
 Proceedings First Annual Conference on Computational Molecular Biology,
 (RECOMB97), ACM Press, 209-215, 1997). However, these heuristics have only
 worked on limited sets of data and their efficacy (or approximability)
 remains unproven.
 SUMMARY OF THE INVENTION
 In accordance with our invention, we use a carefully constructed prior
 model of the cuts to infer the best hypothetical model by using Bayes'
 formula. The solution requires searching over a high-dimensional
 hypothesis space and is complicated by the fact that the underlying
 distributions are multimodal. Nevertheless, the search over this space can
 be accomplished without sacrificing efficiency. Furthermore, one can speed
 up the algorithm by suitably constraining various parameters in the
 implementation (but at the loss of accuracy or an increased probability of
 missing the correct map).
 The main ingredients of this method are the following:
 A model or hypothesis H, of the map of restriction sites.
 A prior distribution of the data in the form of single molecule restriction
 map (SMRM) vectors:
EQU Pr[D.sub.j.vertline.H].
 Assuming pair-wise conditional independence of the SMRM vectors D.sub.j,
EQU Pr[D.sub.j.vertline.D.sub.j1, . . . , D.sub.jm, H]=Pr[D.sub.j.vertline.H].
 Thus, the prior distribution of the entire data set of SMRM vectors becomes
 ##EQU1##
 where index j ranges over the data set.
 A posterior distribution via Bayes' rule
 ##EQU2##
 Here Pr[H] is the unconditional distribution of hypothesis H, and Pr[D] is
 the unconditional distribution of the data.
 Using this formulation, we search over the space of all hypotheses to find
 the most "plausible" hypothesis H that maximizes the posterior
 probability.
 The hypotheses H are modeled by a small number of parameters .PHI.(H)
 (e.g., number of cuts, distributions of the cuts, distributions of the
 false cuts, etc.). We have prior models for only a few of these parameters
 (number of cuts), and the other parameters are implicitly assumed to be
 equi-probable. Thus the model of Pr[H] is rather simplistic. The
 unconditional distribution for the data Pr[D] does not have to be computed
 at all since it does not affect the choice of H. In contrast, we use a
 very detailed model for the conditional distribution for the data given
 the chosen parameter values for the hypothesis. One can write the
 expression for the posterior distribution as:
EQU log(Pr[.PHI.(H).vertline.D])=L+Penalty+Bias,
 where L.tbd..SIGMA..sub.j log (Pr[D.sub.j.vertline..PHI.(H)] is the
 likelihood function, Penalty=log Pr[.PHI.(H)] and Bias=-log(Pr[D])= a
 constant. In these equations .PHI.(H) corresponds to the parameter set
 describing the hypothesis and .PHI. (H).OR right..PHI.(H) a subset of
 parameters that have a nontrivial prior model. In the following, we shall
 often write H for .PHI.(H), when the context creates no ambiguity.
 Also, note that the bias term has no effect as it is a constant
 (independent of the hypothesis), and the penalty term has discernible
 effect only when the data set is small. Thus, our focus is often on the
 term L which dominates all other terms in the right hand side.
 As we will see the posterior distribution, Pr[H.vertline.D] is multimodal
 and the prior distribution Pr[D.sub.j.vertline.H] does not admit a closed
 form evaluation (as it is dependent on the orientation and alignment with
 H). Thus, we need to rely on iterative sampling techniques.
 Thus the search method to find the most "plausible" hypothesis has two
 parts: we take a sample hypothesis and locally search for the most
 plausible hypothesis in its neighborhood using gradient search techniques;
 we use a global search to generate a set of sample hypotheses and filter
 out all but the ones that are likely to be near plausible hypotheses.
 Our approach using this Bayesian scheme enjoys many advantages:
 One obtains the best possible estimate of the map given the data, subject
 only to the comprehensiveness of the model .PHI.(H) used.
 For a comprehensive model H, estimates of .PHI.(H) are unbiased and errors
 converge asymptotically to zero as data size increases.
 Additional sources of error can be modeled simply by adding parameters to
 .PHI.(H).
 Estimates of the errors in the result can be computed in a straightforward
 manner.
 The algorithm provides an easy way to compute a quality measure.

DESCRIPTION OF THE PREFERRED EMBODIMENTS
 This description has several parts. In part A, we describe the restriction
 map model and formulate the underlying algorithmic problems. In part B, we
 describe a statistical model for the problem based on assumptions on the
 distributions of the bases in DNA and the properties of the chemical
 processes involved in optical mapping. These models are then used, in part
 C, to describe a probabilistic local search algorithm with good average
 time complexity. Part D describes an update algorithm which is used in
 conjunction with the local search algorithm to improve computational
 efficiency. Part E describes the global search algorithm, and how it is
 made less computationally expensive through the use of a branch and bound
 algorithm. Finally, part F describes the quality measure used to determine
 the best map.
 FIG. 1 illustrates the global search strategy used in the method of the
 invention. Initially, data is obtained from individual DNA molecules which
 have been digested into fragments by restriction enzymes (Step 500, FIG.
 1). The data (cut sites) is then used to model a restriction map as a
 vector, for each molecule (Step 501, FIG. 1). Next, for each restriction
 map, a hypothesis map which is locally most likely to be the true
 restriction map is found (Step 502, FIG. 1). After the individual locally
 likely hypotheses are ranked according to their likelihood of correctness
 (Step 503, FIG. 1), the most likely hypothesis is chosen as the correct,
 or best, answer (Step 504, FIG. 1).
 FIG. 2 illustrates a method of searching for the locally most likely
 hypotheses (Step 502, FIG. 1) wherein a Bayesian approach is used. A. P.
 Dempster, N. M. Laird and D. B. Rubin, "Maximum Likelihood from Incomplete
 Data via the EM Algorithm," J. Roy. Stat. Soc., 39(1):1-38, 1977; U.
 Grenander and M. I. Miller, "Representations of Knowledge in Complex
 Systems (with discussions)," J. Roy. Stat. Soc. B.,56:549-603. After a
 hypothesis H of the restriction map is selected (Step 511, FIG. 2), a
 prior probability distribution is defined (Step 512, FIG. 2). A posterior
 probability distribution is then defined (Step 513, FIG. 2), which is then
 used to search the space of plausible hypotheses to find the hypothesis
 which is most likely to be correct (that is, the hypothesis which
 maximizes the posterior probability distribution (Step 514, FIG. 2).
 A. Restriction Map Model
 Our problem can be formulated as follows. Assuming that all individual
 single-molecule restriction maps correspond to the same clone, and that
 the imaging algorithm can only provide the fragment size estimates that
 are scaled by some unknown scale factor, we represent a single molecule
 restriction map (SMRM) by a vector with ordered set of rational numbers on
 the open unit interval (0,1):
EQU D.sub.j =(s.sub.1j, s.sub.2j, . . . , s.sub.M.sub..sub.j, .sub.j),
 0&lt;s.sub.1j &lt;s.sub.2j &lt;. . . &lt;s.sub.M.sub..sub.j, .sub.j 1,
 s.sub.ij.epsilon.Q
 Given a rational number s.epsilon.(0,1), its reflection is denoted by
 s.sup.R =1-s. Similarly, by D.sub.j.sup.R, we denote the vector
EQU D.sub.j.sup.R =(s.sub.M.sub..sub.j, .sub.j, . . .
 ,s.sub.2j.sup.R,s.sub.1j.sup.R).
 Note that if the entries of D.sub.j are ordered and belong to the open unit
 interval, so do D.sub.j +c and D.sub.j.sup.R provided that c is
 appropriately constrained.
 Thus our problem can be described as follows:
 Given a collection of SMRM vectors,
EQU D.sub.1,D.sub.2, . . . , Dm
 we need to compute a final vector H=(h.sub.1, h.sub.2, . . . , h.sub.N)
 such that H is "consistent" with each D.sub.j. Thus, H represents the
 correct restriction map and the D.sub.j 's correspond to several
 "corrupted versions" of H. A graphical representation of H and one vector
 D.sub.j, is depicted in FIG. 3.
 We shall define such a general notion of "consistency" using a Bayesian
 approach which depends on the conditional probability that a data item
 D.sub.j can be present given that the correct restriction map for this
 particular clone is H.
 In order to accurately model the prior distribution Pr[D.vertline.H], we
 need to consider the following categories of errors in image data: 1)
 misidentification of spurious materials in the image as DNA, 2)
 identifying multiple DNA molecules as one, 3) identifying partial DNA
 molecules as complete, 4) errors in estimating sizes of DNA fragments, 5)
 incomplete digestion of DNA, 6) cuts visible at locations other than
 digest sites, and 7) unknown orientation of DNA molecule.
 Our prior distribution Pr[D.vertline.H] is modeled as following:
 A molecule on the surface can be read from left to right or right to left.
 The uncertainty in orientation is modeled as Bernoulli processes, with the
 probability for each orientation being equal.
 The restrictions sites on the molecule are determined by a distribution
 induced by the underlying distribution of the four bases in the DNA. For
 example, we shall assume that the probability that a particular base (say,
 A) appears at a location i is independent of the other bases, though the
 probabilities are not necessarily identical.
 The false cuts appear on the molecule as a Poisson process. This is a
 consequence of the simplifying assumption that over a small region
 .DELTA.h on the molecule, the Pr[# False cuts=1 over
 .DELTA.h]=.lambda..sub.f .DELTA.h and the Pr[#False cuts .gtoreq.2 over
 .DELTA.h]=o(.DELTA.h).
 The fragment size (the size of the molecule between two cuts) is estimated
 with some loss of accuracy (dependent on the stretching of the molecule,
 fluorochrome attachments and the image processing algorithm). The measured
 size is assumed to be distributed as a Gaussian distribution.
 The following notation will be used to describe the parameters of the
 independent processes responsible for the statistical structure of the
 data. Unless otherwise specified, the indices i, j and k are to have the
 following interpretation:
 The index i ranges from 1 to N and refers to cuts in the hypothesis.
 The index j ranges from 1 to M and refers to data items (i.e., molecules).
 The index k ranges from 1 to K and refers to a specific alignment of cuts
 in the hypothesis vs. data.
 Now the main parameters of our Bayesian model are as follows:
 P.sub.ci =Probability that the ith sequence specific restriction site in
 the molecule will be visible as a cut.
 .sigma..sub.i =Standard deviation of the observed position of the ith cut
 when present and depends on the accuracy with which a fragment can be
 sized.
 .lambda..sub.f =Expected number of false-cuts per molecule observed. Since
 all sizes will be normalized by the molecule size, this will also be the
 false-cuts per unit length.
 P.sub.b =Probability that the data is invalid ("bad"). In this case, the
 data item is assumed to have no relation to the hypothesis being tested,
 and could be an unrelated piece of DNA or a partial molecule with a
 significant fraction of the DNA missing. The cut-sites (all false) on this
 data item are assumed to have been generated by a Poisson process with the
 expected number of cuts =.lambda..sub.n.
 Note that the regular DNA model reduces to the "bad" DNA model for the
 degenerate situation when p.sub.ci.fwdarw.0 and
 .lambda..sub.f.fwdarw..lambda..sub.n. As a result, "bad" DNA molecules
 cannot be disambiguated from regular DNA molecules if p.sub.ci.apprxeq.0.
 In practice, p.sub.ci &gt;0 and .lambda..sub.n &gt;.lambda..sub.f, and the
 degenerate case almost never occurs. Here the "bad" molecules are
 recognized by having a disproportionately large number of false cuts.
 .lambda..sub.n =Expected number of cuts per "bad" molecule.
 B. Prior Distribution in the Hypotheses Space
 Recall that by Bayes' rule
 ##EQU3##
 Assuming that the prior distribution Pr[Hr] is given in terms of just the
 number of restriction sites, based on the standard Poisson distribution,
 we wish to find the "most plausible" hypothesis JHby maximizing
 Pr[H.vertline.D].
 In our case, H is simply the final map (a sequence of restriction sites,
 h.sub.1, h.sub.2, . . . h.sub.N) augmented by the auxiliary parameters
 such as P.sub.ci, .sigma..sub.i, .lambda..sub.f, etc. When we compare a
 data item D.sub.j with respect to this hypothesis, we need to consider
 every possible way that D.sub.j could have been generated by H. In
 particular we need to consider every possible alignment, where the kth
 alignment, A.sub.jk, corresponds to a choice of the orientation for
 D.sub.j as well as identifying a cut on D.sub.j with a true restriction
 site on H or labeling the cut as a false cut. By D.sub.j.sup.(Ajk) [also
 abbreviated as D.sub.j.sup.(k) ], we shall denote the "interpretation of
 the jth data item with respect to the alignment A.sub.jk." Each alignment
 describes an independent process by which D.sub.j could have been
 generated from H, and therefore the total probability distribution of
 D.sub.j is the sum of the probability distribution of all these
 alignments, plus the remaining possible derivations (invalid data).
 As a consequence of the pairwise independence and the preceding discussion,
 we have the following:
 ##EQU4##
 where index j ranges over the data set, and
 ##EQU5##
 where index k ranges over the set of alignments.
 In the preceding equation, Pr[D.sub.j.sup.(k).vertline.H,
 good](abbreviated, Pr.sub.jk) is the probability distribution of model
 D.sub.j being derived from model H and corresponding to a particular
 alignment of cuts (denoted, A.sub.jk). The set of alignments includes
 alignments for both orientations, hence each alignment has a prior
 probability of 1/2. If D.sub.j is bad, our model corresponds to H with
 p.sub.ci.fwdarw.0 and .lambda..sub.f.fwdarw..lambda..sub.n. We shall often
 omit the qualifier "good" for the hypothesis H, when it is clear from the
 context.
 Thus, in the example shown in FIG. 4, for a given hypothesis H, the
 conditional probability distribution that the jth data item D.sub.j with
 respect to alignment A.sub.jk (i.e., D.sub.j.sup.(k)) could have occurred
 is given by the following formula:
 ##EQU6##
 In the most general case, we proceed as follows. Let
 N.tbd.Number of cuts in the hypothesis H.
 h.sub.i.tbd.The ith cut location on H.
 M.sub.j.tbd.Number of cuts in the data D.sub.j.
 K.sub.j.tbd.Number of possible alignments of the data/evidence D.sub.j
 against the hypothesis H (or its reversal, the flipped alignment H.sup.R).
 S.sub.ijk.tbd.The cut location in D.sub.j matching the cut h.sub.i in H,
 given the alignment A.sub.jk. In case such a match occurs, this event is
 denoted by an indicator variable m.sub.ijk taking the value 1.
 m.sub.ijk.tbd.An indicator variable, taking the value 1 if and only if the
 cut S.sub.ijk in D.sub.j matches a cut h.sub.i in the hypothesis H, given
 the alignment A.sub.jk. It takes the value 0, otherwise.
 F.sub.jk.tbd.Number of false (non-matching) cuts in the data D.sub.j for
 alignment A.sub.jk, that do not match any cut in the hypothesis H. Thus
 ##EQU7##
 The number of missing cuts is thus
 ##EQU8##
 We may omit the indices j and k, if from the context it can be uniquely
 determined which data D.sub.j and alignment A.sub.jk are being referred
 to.
 Note that a fixed alignment A.sub.jk can be uniquely described by marking
 the cuts on D.sub.j by the labels T (for true cut) and F (for false cut)
 and by further augmenting each true cut by the identity of the cut h.sub.i
 of the hypothesis H. From this information, m.sub.ijk, S.sub.ijk,
 F.sub.jk, etc. can all be uniquely determined. Let the cuts of D.sub.j be
 (S.sub.1, S.sub.2, . . . , S.sub.Mj). Let the event E.sub.i denote the
 situation that there is a cut in the infinitesimal interval (s.sub.i
 -.DELTA.x/2, s.sub.i +.DELTA.x/2). Thus we have:
 ##EQU9##
 Note the following:
 ##EQU10##
 For the event E.sub..DELTA. there are two possible situations to be
 considered:
 (a) s.sub..DELTA. is a false cut and the number of false cuts among
 s.sub.1, . . . , s.sub..DELTA.-1 is .beta..
EQU prob[E.sub..DELTA., A.sub.jk.vertline.E.sub.1, . . . , E.sub..DELTA.-1,
 m.sub.ijk, M.sub.j, H, good]=(F.sub.jk -.beta.).DELTA.x.
 (b) s.sub..DELTA. =S.sub.ijk is a true cut and h.sub.i is the cut in H
 associated with it.
 ##EQU11##
 Thus,
 ##EQU12##
 Putting it all together:
 ##EQU13##
 By an identical argument we see that the only alignments relevant for the
 bad molecules correspond to the situation when all cuts in D.sub.j are
 labeled false, and for each of two such alignments,
EQU Pr[D.sub.j.sup.(k).vertline.H,bad]=e.sup.-.lambda..sup..sub.n
 .lambda..sub.n.sup.M.sup..sub.j .
 The log-likelihood can then be computed as follows:
 ##EQU14##
 Thus,
 ##EQU15##
 In the model, we shall use an extremely simple distribution on the prior
 distribution Pr[H] that only depends on the number of restriction sites N
 and not any other parameters. Implicitly, we assume that all hypotheses
 with the same number of cuts are equi-probable, independent of the cut
 location.
 Given a k-cutter enzyme (e.g., normally six-cutters like EcoR I in our
 case), the probability that it cuts at any specific site in a sufficiently
 long clone is given by
 ##EQU16##
 Thus if the clone is of length G bps we denote by .lambda..sub.e =GP.sub.e
 (the expected number of restriction sites in the clone), then the
 probability that the clone has exactly N restriction cuts is given by
 ##EQU17##
 The preceding computation is based on the assumption that all four bases
 .epsilon.{(A,T,C,G} occur in the clone with equal probability 1/4.
 However, as is known, the human genome is CG-poor (i.e.,
 Pr[C]+Pr[G]=0.32&lt;Pr[A]+P[T]=0.68). D. Barker, M. Schafer and R. White,
 "Restriction Sites Containing CpG Show a Higher Frequency of polymorphism
 in Human DNA," Cell, 36:131-138, (1984). A more realistic model can use a
 better estimation for P.sub.e :
EQU P.sub.e =(0.16).sup.#CG (0.34).sup.#AT,
 where #CG denotes the number of C or G in the restriction sequence for the
 enzyme and, similarly, #AT denotes the number of A or T in the restriction
 sequence.
 C. Local Search Algorithm
 In order to find the most plausible restriction map, we shall optimize the
 cost function derived earlier, with respect to the following parameters:
 Cut Sites=h.sub.1, h.sub.2, . . . , h.sub.N,
 Cut Rates=p.sub.c1, p.sub.c2, . . . , p.sub.cn,
 Std. Dev. of Cut Sites=.sigma..sub.1, .sigma..sub.2, . . . ,.sigma..sub.N,
 Auxiliary Parameters=p.sub.b, .lambda..sub.f and .lambda..sub.n.
 Let us denote any of these parameters by .theta.. Thus, we need to solve
 the equation
 ##EQU18##
 to find an extremal point of L with respect to the parameter .theta..
 Case 1: .theta.=p.sub.b
 Taking the first partial derivative, we get
 ##EQU19##
 Taking the second partial derivative, we get
 ##EQU20##
 Accordingly, L can now be easily optimized iteratively to estimate the best
 value of p.sub.b, by means of the following applications of the Newton's
 equation:
 ##EQU21##
 Case 2: .theta.=.lambda..sub.n
 This parameter is simply estimated to be the average number of cuts. Note
 that
 ##EQU22##
 should be zero at the local maxima. Thus a good approximation is obtained
 by taking
 ##EQU23##
 leading to the update rule
 ##EQU24##
 Thus .lambda..sub.n is simply the average number of cuts per molecule.
 Case 3: .theta.=h.sub.i, p.sub.ci, .sigma..sub.i (i=1, . . . , N), or
 .lambda..sub.f
 Unlike in the previous two cases, these parameters are in the innermost
 section of our probability distribution expression and computing any of
 these gradients will turn out to be computationally comparable to
 evaluating the entire probability distribution.
 In this case,
 ##EQU25##
 where
 ##EQU26##
 For convenience, now define
 ##EQU27##
 .tbd. Relative probability distribution of the alignment A.sub.jk for data
 item D.sub.j. Thus, our earlier formula for
 ##EQU28##
 now simplifies to
 ##EQU29##
 Before examining the updating formula for each parameter optimization, we
 shall introduce the following notations for future use. The quantities
 defined below can be efficiently accumulated for a fixed value of the set
 of parameters.
 .PSI..sub.0i.tbd..SIGMA..sub.j.SIGMA..sub.k.pi..sub.jk
 M.sub.ijk.tbd.Expected number of cuts matching h.sub.i
 .PSI..sub.1i.tbd..SIGMA..sub.j.SIGMA..sub.k.pi..sub.jk M.sub.ijk
 S.sub.ijk.tbd.Sum of cut locations matching h.sub.i
 .PSI..sub.2i.tbd..SIGMA..sub.j.SIGMA..sub.k.pi..sub.jk M.sub.ijk
 S.sub.ijk.sup.2.tbd.Sum of square of cut locations matching h.sub.i
 .mu..sub.g.tbd..SIGMA..sub.j.SIGMA..sub.k.pi..sub.jk.tbd.Expected number of
 "good" molecules
 .gamma..sub.g.tbd..SIGMA..sub.j.SIGMA..sub.k.pi..sub.jk
 M.sub.j.tbd.Expected number of cuts in "good" molecules
 We note here that the .PSI.'s can all be computed efficiently using a
 simple updating rule that modifies the values with one data item D.sub.j
 (molecule) at a time. This rule can then be implemented using a Dynamic
 Programming recurrence equation (described later).
 Case 3A: .theta.=h.sub.i Note that
 ##EQU30##
 Thus,
 ##EQU31##
 Although the .PSI.'s depend on the location h.sub.i, they vary rather
 slowly as a function of h.sub.i. Hence a feasible update rule for h.sub.i
 is
 ##EQU32##
 Thus the updated value of h.sub.i is simply the "average expected value" of
 all the S.sub.ijk 's that match the current value of h.sub.i.
 Case 3B: .theta.=p.sub.ci Note that
 ##EQU33##
 Thus,
 ##EQU34##
 Again, arguing as before, we have the following feasible update rule for
 p.sub.ci
 ##EQU35##
 Thus p.sub.ci is just the fraction of the good molecules that have a
 matching cut at the current value of h.sub.i.
 Case 3C: .theta.=.sigma..sub.i Note that
 ##EQU36##
 Thus,
 ##EQU37##
 Thus, we have the following feasible update rule for .sigma..sub.i.sup.2
 ##EQU38##
 Using the estimate for h.sub.i (equation (4)), we have
EQU .sigma..sub.i.sup.2 :=.PSI..sub.2i /.PSI..sub.0i -(.PSI..sub.1i
 /.PSI..sub.0i).sup.2 (6)
 This is simply the variance of all the s.sub.ijk 's that match the current
 value of h.sub.i.
 Case 3D: .theta.=.lambda..sub.f Note that
 ##EQU39##
 Thus, we have the following feasible update rule for .lambda..sub.f
 ##EQU40##
 This is simply the average number of unmatched cuts per "good" molecule.
 (Note that the molecules are already normalized to unit length.)
 Case 3E: .theta.=p.sub.c =p.sub.c1 =. . . =p.sub.cN (Constrained) Note that
 ##EQU41##
 Thus the update equation for this case is:
 ##EQU42##
 Case 3F: .theta.=.sigma.=.sigma..sub.1 =. . . =.sigma..sub.N (Constrained)
 Note that
 ##EQU43##
 Thus the update equation for this case is:
 ##EQU44##
 D. Update Algorithm: Dynamic Programming
 In each update step, we need to compute the new values of the parameters
 based on the old values of the parameters, which affect the "moment
 functions": .PSI..sub.0i, .PSI..sub.1i, .PSI..sub.2i, .mu..sub.g and
 .gamma..sub.g. For the ease of expressing the computation, we shall use
 additional auxiliary expressions as follows:
 ##EQU45##
 One motivation for this formulation is to avoid having to compute
 e.sup.-.lambda.f repeatedly, since this is a relatively expensive
 computation. Note that the original moment function can now be computed as
 follows:
 ##EQU46##
 Finally,
 ##EQU47##
 The definitions for P.sub.j, W.sub.ij, SUM.sub.ij and SQ.sub.ij involve all
 alignments between each data element D.sub.j and the hypothesis H. This
 number is easily seen to be exponential in the number of cuts N in the
 hypothesis H, even if one excludes such physically impossible alignments
 as the ones involving cross-overs (i.e., alignments in which the order of
 cuts in H and D.sub.j are different). First, consider P.sub.j :
 ##EQU48##
 Next we shall describe recurrence equations for computing the values for
 all alignments efficiently. The set of alignments computed are for the
 cuts {1, . . . , M.sub.j } of D.sub.j mapped against the hypothesized cuts
 {1, . . . , N}. We define the recurrence equations in terms of
EQU P.sub.q,r.tbd.P.sub.j (s.sub.q, . . . , s.sub.M.sub..sub.j ;h.sub.r, . . .
 , h.sub.N),
 which is the probability density of all alignments for the simpler problem
 in which cuts s.sub.1, . . . , s.sub.q-1 are missing in the data D.sub.j
 and the cuts h.sub.1, . . . , h.sub.r-1 are missing in the hypothesis H.
 Then, clearly
 ##EQU49##
 This follows from a nested enumeration of all possible alignments. The
 recurrence terminates in P.sub.Mj+1,r, which represents P.sub.j if all
 cuts in D.sub.j were missing and cuts h.sub.1, . . . , h.sub.r-1 in H were
 missing:
 ##EQU50##
 where 1.ltoreq.q.ltoreq.M.sub.j and 1&lt;r&lt;N+1.
 Thus the total number of terms P.sub.q,r to be computed is bounded from
 above by (M.sub.j +1) (N+1) where M.sub.j is the number of cuts in data
 molecule D.sub.j and N is the number of cuts in H. Each term can be
 computed in descending order of q and r using equations (12) and (13). The
 time complexity associated with the computation of P.sub.q,r is O(N-r) in
 terms of the arithmetic operations.
 Note also that the equation (12) can be written in the following
 alternative form:
 ##EQU51##
 where 1.ltoreq.q.ltoreq.M.sub.j and 1.ltoreq.r.ltoreq.N+1.
 Thus, by computing P.sub.q,r in descending order of r, only two new terms
 [and one new product (1-p.sub.cr) in equation (14)] need be to be computed
 for each P.sub.q,r. With this modification, the overall time complexity
 reduces to O(M.sub.j N).
 The complexity can be further reduced by taking advantage of the fact that
 the exponential term is negligibly small unless h.sub.t and s.sub.q are
 sufficiently close (e.g., .vertline.h.sub.t
 -s.sub.q.vertline..ltoreq.3.sigma..sub.t). For any given value of q, only
 a small number of h.sub.t will be close to s.sub.q. For a desired finite
 precision only a small constant fraction of h.sub.t 's will be
 sufficiently close to s.sub.q to require that the term with the exponent
 be included in the summation (in practice, even a precision of 10.sup.-10
 will only require 3-5 terms to be included with a around 1%).
 Note however that even with this optimization in the computation for
 equation (12), the computation of P.sub.q,r achieves no asymptotic
 improvement in the time complexity, since P.sub.q,r with consecutive r can
 be computed with only two new terms, as noted earlier. However, for any
 given q, only for a few r values are both of these additional terms
 non-negligible. The range of r values (say, between r.sub.min and
 r.sub.max) for which the new terms with e.sup.-(hr-sq).sup..sup.2
 .sup./2.sigma..sup..sub.t .sup..sup.2 are significant can be precomputed
 in a table indexed by q=1, . . . , M.sub.j. For r&gt;r.sub.max all terms in
 the summation are negligible. For r&lt;r.sub.min the new exponential term
 referred to previously is negligible. In both cases, the expression for
 P.sub.q,r can be simplified:
 ##EQU52##
 Since both r.sub.min [q] and r.sub.max [q] are monotonically nondecreasing
 functions of q, the (q,r) space divides as shown in FIG. 5. Of course, the
 block diagonal pattern need not be as regular as shown and will differ for
 each data molecule D.sub.j.
 Note again that our ultimate object is to compute P.sub.1,1. Terms
 P.sub.q,r+1 with r&gt;r.sub.max [q], cannot influence any term P.sub.q',r'
 with r'.ltoreq.r (see equation (12)). Therefore, any term P.sub.q, r+1
 with r&gt;r.sub.max [q] cannot influence P.sub.1,1 as is readily seen by a
 straightforward inductive argument. Therefore, all such terms need not be
 computed at all.
 For r&lt;r.sub.min [q], these terms are required but need not be computed
 since they always satisfy the following identity:
EQU P.sub.q,r =(1-p.sub.c.sub..sub.r )P.sub.q,r+1, r&lt;r.sub.min [q].
 This follows from equations (13) and (15) by induction on q. These terms
 can then be generated on demand when the normal recurrence (equation (12))
 is computed and whenever a term P.sub.q+1r, is required for which
 r&lt;r.sub.min [q+1], provided terms are processed in descending order of r.
 Thus, the effective complexity of the algorithm is O(M.sub.j (r.sub.max
 -r.sub.min +2)). Since r.sub.max -r.sub.min +2 is proportional for a given
 precision to .left brkt-top.(.sigma.N+1).right brkt-top., (where .sigma.
 is an upper bound on all the .sigma..sub.t values) we see that the time
 complexity for a single molecule D.sub.j is O(.sigma.M.sub.j N). Summing
 over all molecules D.sub.j the total time complexity is O(.sigma.MN),
 where M=.SIGMA..sub.j M.sub.j. The space complexity is trivially bounded
 by O(M.sub.max N) where M.sub.max max.sub.j M.sub.j.
 Essentially the same recurrence equations can be used to compute W.sub.ij,
 SUM.sub.ij and SQ.sub.ij, since these 3N quantities sum up the same
 probability distributions Pr.sub.jk weighted by m.sub.ijk, m.sub.ijk
 S.sub.ijk or m.sub.ijk S.sub.ijk.sup.2, respectively. The difference is
 that the termination of the recurrence (cf equation (13)) is simply
 P.sub.M.sub..sub.j .sub.+1,r =0, whereas the basic recurrence equation (cf
 equation(12)) contains an additional term corresponding to the m.sub.ijk
 times the corresponding term in the recurrence equation. For example:
 ##EQU53##
 where 1.ltoreq.q .ltoreq.M.sub.j and 1.ltoreq.r.ltoreq.N+1.
 Note that the new term is only present and as before need only be computed
 if the corresponding exponent is significant, i.e., i lies between
 r.sub.min [q] and r.sub.max [q]. This term is the only nonzero input term
 in the recurrence since the terminal terms are zero. This recurrence is
 most easily derived by noting (from equations (1) and (10)) that the sum
 of products form of SUMS.sub.ij can be derived from that of P.sub.j by
 multiplying each product term with h.sub.i -s.sub.q in any exponent by
 S.sub.q and deleting any term without h.sub.i in the exponent. Since each
 product term contains at most one exponent with h.sub.i this
 transformation can also be applied to the recurrence form for P.sub.j
 (equation (12)), which is just a different factorization of the original
 sum of products form. The result is equation (16). The corresponding
 derivation for W.sub.ij and SQ.sub.ij is the same except that the s.sub.q
 is replaced by 1 or s.sub.q.sup.2 respectively. If the recurrences for
 these 3N quantities are computed in parallel with the probability
 distribution P.sub.j, the cost of the extra term is negligible, so the
 overall cost of computing both the probability distribution P.sub.j and
 its gradients is O(.sigma.MN.sup.2). The cost of conversion equations (11)
 is also negligible in comparison. Moreover this can be implemented as a
 vectorized version of the basic recurrence with vector size 3N+1 to take
 advantage of either vector processors or superscalar pipelined processors.
 We note in passing that if 3N is significantly greater than the average
 width .sigma.M of the dynamic programming block diagonal matrix shown in
 FIG. 3, then a standard strength reduction can be applied to the
 vectorized recurrence equations trading the 3N vector size for a
 .sigma.N+1 vector size and resulting in an alternate complexity of
 O(.sigma..sup.2 MN.sup.2). Note that the gradient must be computed a
 number of times (typically 10-20 times) for the parameters to converge to
 a local maximum and the gain is significant only when .sigma.&lt;&lt;1.
 E. Global Search Algorithm
 Recall that our prior distribution Pr[D.vertline.H] is multimodal and the
 local search based on the gradients by itself cannot evaluate the best
 value of the parameters. Instead, we must rely on some sampling method to
 find points in the parameter space that are likely to be near the global
 maxima. Furthermore, examining the parameter space, we notice that the
 parameters corresponding to the number and locations of restriction sites
 present the largest amount of multimodal variability and hence the
 sampling may be restricted to just h=(N; h.sub.1, h.sub.2, . . . ,
 h.sub.N). The conditional observation probability distribution
 Pr[D.vertline.H] can be evaluated pointwise in time O(.sigma.MN) and the
 nearest local maxima located in time O(.sigma.MN.sup.2), though there is
 no efficient way to sample all local maxima exhaustively.
 Thus, our global search algorithm will proceed as follows: we shall first
 generate a set of samples (h.sub.1, h.sub.2, h.sub.3, . . . ); these
 points are then used to begin a gradient search for the nearest maxima and
 provide hypotheses (H.sub.1, H.sub.2, H.sub.3,. . . ); the hypotheses are
 then ranked in terms of their posterior probability distribution
 Pr[H.vertline.D] (whose relative values also lead to the quality measure
 for each hypothesis) and the one (or more) leading to maximal posterior
 probability distribution is presented as the final answer.
 However, even after restricting the sampling space, the high dimension of
 the space makes the sampling task daunting. Even if the space is
 discretized (for instance, each h.sub.i .epsilon.{0, 1/200, . . . , j/200,
 . . . , 1}), there are still far too many sample points (200.sup.N) even
 for a small number of cuts (say, N=8). However, the efficiency can be
 improved if we accept an approximate solution.
 We use approximate Bayesian probability distributions in conjunction with a
 branch and bound algorithm to reject a large fraction of the samples
 without further local analysis.
 For the present the parameter N is searched in strictly ascending order.
 This means one first evaluates the (single) map with no cuts, then applies
 global and gradient search to locate the best map with 1 cut, then the
 best map with 2 cuts etc. One continues until the score of the best map of
 N cuts is significantly worse than the best map of 0 . . . N-1 cuts.
 The global search for a particular N uses an approximate Bayesian
 probability distribution with a scoring function that is amenable to an
 efficient branch-and-bound search. Observe that obtaining good scores for
 some molecule D.sub.j basically requires that as many cut locations
 S.sub.1j, . . . , S.sub.Mj,j as possible must line up close to h.sub.1,
 h.sub.2, . . . , h.sub.N in one of the two orientations. This means that
 any subset of the true map h.sub.1, h.sub.2, . . . , h.sub.m (m&lt;N) will
 score better than most other maps of size m, assuming that the digestion
 rate is equal (p.sub.c =p.sub.ci =. . . =p.sub.cN). Note that for physical
 reasons, the variation among the digestion rates is quite small; thus, our
 assumption is valid and permits us to explicitly constrain these
 parameters to be the same. For example, if (h.sub.1, h.sub.2, . . . ,
 h.sub.N) is the correct map, one expects maps with single cuts located at
 [h.sub.i ](1.ltoreq.i.ltoreq.N) to score about equally well in terms of
 the Bayesian probability distribution. Similarly, maps with two cuts
 located at pairs of [h.sub.i, h.sub.j ](1.ltoreq.i&lt;j.ltoreq.N) score about
 equally well and better than arbitrarily chosen two cut maps. Furthermore,
 the pair-cut probability distribution are more robust than the single cut
 probability distribution with respect to the presence of false cuts,
 hence, less likely to score maps with cuts in other than the correct
 locations. Hence an approximate score function used for a map (h.sub.1,
 h.sub.2, . . . , h.sub.N) is the smallest probability distribution for any
 pair map [h.sub.i, h.sub.i ] which is a subset of (h.sub.1, h.sub.2, . . .
 , h.sub.N). These pair map probability distribution can be precomputed for
 every possible pair ([h.sub.i, h.sub.j ]) if h.sub.i, h.sub.j are forced
 to have only K values along some finite sized grid, for example at exact
 multiples of 1/2% of the total molecule length for K=200. The pair map
 probability distribution can then be expressed in the form of a complete
 undirected graph, with K nodes corresponding to possible locations, and
 each edge between node i to j having an edge value equal to the
 precomputed pair map probability distribution of [h.sub.i, h.sub.j ]. A
 candidate map (h.sub.1, h.sub.2, . . . , h.sub.N) corresponds to a clique
 of size N in the graph, and its approximate score corresponds to the
 smallest edge weight in the clique.
 In general, the clique problem (for instance, with binary edge weights) is
 NP-complete and may not result in any asymptotic speedup over the
 exhaustive search. However, for our problem effective branch-and-bound
 search heuristics can be devised. Consider first the problem of finding
 just the best clique. We can devise two bounds that can eliminate much of
 the search space for the best clique:
 The score of any edge of a clique is an upper bound on the score of that
 clique. If the previous best clique found during a search has a better
 (higher) score than the score of some edge, all cliques that include this
 edge can be ruled out.
 For each node in the graph, one can precompute the score of the best edge
 that includes this node. If the previous best clique found during a search
 has a better (higher) score than this node score, all cliques that include
 this node can be ruled out.
 As with all branch-and-bound heuristics, the effectiveness depends on
 quickly finding some good solutions, in this case cliques with good
 scores. We have found that an effective way is to sort all K nodes by the
 Bayesian scores of the corresponding single cut map. In other words, we
 first try nodes that correspond to restriction site locations that have a
 high observed cut rate in some orientation of the molecules. Also the
 nodes corresponding to cut sites of the best overall map so far (with
 fewer than N cut sites) are tried first.
 For data consisting of a few hundred molecules, the branch-and-bound
 heuristics allows exhaustive search in under a minute on a Sparc System 20
 with N.ltoreq.7 (with K=200). For N&gt;7, a simple step wise search procedure
 that searches for the best map (h.sub.1, h.sub.2, . . . , h.sub.n) by
 fixing N-7 nodes based on the previous best map, works well. The N-7 nodes
 selected are the optimal with respect to a simple metric, for instance,
 the nodes with the smallest standard error (i.e., ratio of standard
 deviation to square root of sample size).
 Next, the global search is modified to save the best B (typically 8000)
 cliques of each size and then the exact Bayesian probability distribution
 is evaluated at each of these B locations, adding reasonable values for
 parameters other than (N; h.sub.1, . . . , h.sub.N). These parameters can
 be taken from the previous best map, or by using some prior values if no
 previous best map is available. For some best scoring subset (typically
 32-64) of these maps gradient search is used to locate the nearest maxima
 (and also accurate estimates for all parameters), and the best scoring
 maxima is used as the final estimate for the global maxima for the current
 value of N.
 The branch-and-bound heuristics was modified to find the best B cliques, by
 maintaining the best B cliques (found so far) in a priority queue (with an
 ordering based on the approximate score).
 F. A Ouality Measure for the Best Map
 As a quality measure for the best map, we use the estimated probability of
 the dominant mode (peak) of the posterior probability distribution. This
 could be computed by integrating the probability distribution over a small
 neighborhood of the peak (computed in the parameter space). Our cost
 function corresponds to a constant multiple of the posterior probability
 distribution, as we do not explicitly normalize the cost function by
 dividing by a denominator corresponding to the integral of the cost over
 the entire parameter space. To compute the quality measure we make the
 following simplifying assumption: "All peaks are sharp and the integral of
 the cost function over a neighborhood where the cost value is larger than
 a specific amount is proportional to the peak distribution." Also if we
 know the N most dominant peaks (typically N=64), we can approximate the
 integral over all space, by the integral over the N neighborhoods of these
 peaks. Thus we estimate our quality measure for the best map by the ratio
 of the value assigned to it (the integral of the cost function in a small
 neighborhood around it) to the sum of the values assigned to the N best
 peaks. This, of course, simplifies the computation while producing a
 rather good estimate. To take into account the sampling errors (when the
 number of molecules is small) we penalize (reduce) the distribution of the
 best map by an estimate of the sampling error. This approach makes the
 computed quality measure somewhat pessimistic but provides a lower bound.
 As will be apparent to those skilled in the art, numerous modifications may
 be made to the specific embodiment of the invention described above that
 are within the spirit and scope of the invention.