A layout of five clones in a contig based on their actual genome coordinates 

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... algorithm is used to partition it by removing the weakest set of edges (i.e., minimum total edge weight) [11]. After a component is partitioned into two subgraphs, we check both subgraphs against the three conditions. If at least one is satisfied, we partition that component again. This iterative process terminates as soon as there are no more components that need to be split. Elimination of spurious components. In order to introduce the notion of spurious components, let us assume for the time being that the exact locations of the clones in a contig are known. Let U be a connected component of G , and let N ( U ) be the corresponding connected component cloneset. We call U spurious if the overlapping region for the clones in N ( U ) is spanned by another clone in the contig. An example is illustrated in the Appendix. This step is crucial to reduce the number of redundant clones in the preliminary MTP produced by MTP-ILP. Recall that MTP-ILP selects the smallest set of clones that cover each connected component in the MFG. If the spurious components are not removed, at least one clone in each spurious component is added to the MTP without contributing to the overall coverage. Since the exact ordering of the clones is not available, we employ a probabilistic method to detect spurious components. More specifically, we compute the probability that a connected component is observed purely by chance. If this probability is high then we mark the component as spurious. We expect the number of spurious components to be low, since missing fragments or extra fragments are unlikely to occur frequently for a specific cloneset. In our model, we assume that fragment sizes are distributed uniformly, and that the probability that two clones share a fragment is P = 2 T / gellen [19], where T is the tolerance and gellen is the number of possible fragment size values. The probability that two clones share f fragments is P f and the probability that c clones share f fragments is P ( c − 1) f . This suggests that the probability that a connected component U is observed Removing clones that are buried into multiple clones. Recall that the goal of the first step in the construction of the MFG is to bury clones into other clones to reduce the problem size. In this step, we reduce the problem size further by removing clones that are buried into multiple clones. For example, in Figure 3 in the Appendix, clone C is buried into B ∪ D , thus C can be removed. For each clone in a contig, we compute the ratio between the number of its fragments in the MFG and the total number of its fragments. If at least B % of its fragments are already in the MFG, then we mark that clone buried. However, there is a complication. It is possible that two clones are mutually buried and therefore we cannot remove both. In order to solve this problem, we first identify all possible candidates that can be buried into multiple clones. Then, we examine each connected component U of G , and save N ( U ) in a list L if all clones in N ( U ) are candidates to be buried. All candidates that do not exist in any N ( U ) ∈ L can be buried. However, we need to make sure that at least one clone in each N ( U ) ∈ L is not buried, otherwise the region in the target genome that is represented by U is not covered. So the task at hand is to remove as many candidate clones as possible with the constraint that at least one clone survives from each cloneset N ( U ) ∈ L . This problem is called minimum hitting set and it is NP-complete (polynomial reduction from the vertex cover problem [8]). Here, we solve it sub-optimally using a greedy approach [8]. At each iteration, we (1) select the clone c that occurs in the maximum number of clone sets in L , (2) remove all clone sets that contain c , (3) save c into minimum hitting set H , and (4) repeat. The iterative process terminates when the list L is empty. Buried clones selected in this way are removed from the MFG permanently. If this removal introduces components with only one node, they are also removed from the MFG permanently. The algorithm is summarized as Algorithm 1 in the Appendix. in Removing Adding the MFG mandatory clones is P that clones , are where buried to the f is MFG. into number multiple Clones of times clones. that that have Recall N to ( U be ) forms selected that the a connected goal as MTP of the clones com- first ponent. step are called in the In mandatory our construction algorithm, clones. of if the ( | U More MFG | − 1) specifically, f is is to lower bury than clones if at a least threshold into 50% other Q of clones then the fragments we to mark reduce U of the as a spurious. problem clone are size. Spurious not In present this components step, in the we MFG, reduce are removed then the problem that from clone the size is MFG further marked permanently. by as removing mandatory. clones When that a are clone buried is marked into multiple as mandatory, clones. it For is example, immediately in Figure 3 stored in in the the MTP Appendix, and ignored clone C for is buried further into analysis B ∪ D by , thus removing C can be all removed. of its fragments from the MFG. For each clone in a contig, we compute the ratio between the number of its fragments in the MFG and the total number of its fragments. If at least B % of its fragments are already in the MFG, then we mark that clone buried. However, there is a complication. It is possible that two clones are mutually buried and therefore we cannot remove both. In order to solve this problem, we first identify all possible candidates that can be buried into multiple clones. Then, we examine each connected component U of G , and save N ( U ) in a list L if all clones in N ( U ) are candidates to be buried. All candidates that do not exist in any N ( U ) ∈ L can be buried. However, we need to make sure that at least one clone in each N ( U ) ∈ L is not buried, otherwise the region in the target genome that is represented by U is not covered. So the task at hand is to remove as many candidate clones as possible with the constraint that at least one clone survives from each cloneset N ( U ) ∈ L . This problem is called minimum hitting set and it is NP-complete (polynomial reduction from the vertex cover problem [8]). Here, we solve it sub-optimally using a greedy approach [8]. At each iteration, we (1) select the clone c that occurs in the maximum number of clone sets in L , (2) remove all clone sets that contain c , (3) save c into minimum hitting set H , and (4) repeat. The iterative process terminates when the list L is empty. Buried clones selected in this way are removed from the MFG permanently. If this removal introduces components with only one node, they are also removed from the MFG permanently. The algorithm is summarized as Algorithm 1 in the Appendix. Adding mandatory clones to the MFG. Clones that have to be selected as MTP clones are called mandatory clones. More specifically, if at least 50% of the fragments of a clone are not present in the MFG, then that clone is marked as mandatory. When a clone is marked as mandatory, it is immediately stored in the MTP and ignored for further analysis by removing all of its fragments from the MFG. Recall that the MTP can be computed by selecting the smallest set of clones that cover all connected components of the MFG. This problem is a special case of the minimum hitting set problem. Although this problem is NP-complete on general graphs [8], it can be solved optimally in polynomial time on overlap (or interval) graphs [3]. Here, we solve this problem optimally by expressing it as an integer linear program (ILP), but first we remove any connected component from the MFG that does not a ff ect the solution. We reduce the problem size by removing some of the connected components of G according to the following Lemma (the proof is immediate). This simplification step dras- tically reduces the execution time required to solve the ILP (about 200-fold for ...

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... to our experiments, B should be at least 80 to avoid false positive buried clones. If two clones can be buried into each other, the smaller of the two (i.e., the one with fewer fragments) is buried into the other one. FPC also buries clones during the process of building the physical map, but it does not discard them during MTP computation. Building the preliminary MFG. First, we align the fingerprint data for each pair of overlapping clones. For each clone pair ( c i , c j ) for which S ( c i , c j ) ≤ C , we build a bipartite graph G i , j = ( L i ∪ R j , E i , j ), where L i and R j consist of the fragments of c i and c j , respectively, and E i , j = { ( u , v ) | u ∈ L i , v ∈ R j such that | b ( u ) − b ( v ) | ≤ T } . In order to align clones c i and c j , we search for the maximum bipartite matching in G i , j . The matching of maximum cardinality is found by solving max flow on the corresponding flow network [6]. Let M i , j be set of matched edges. For all clone pairs c i and c j for which S ( c i , c j ) ≤ C , the matching edges in M i , j are used to create the (preliminary) matching fragment graph G . Specifically, for each edge ( u , v ) ∈ M i , j , nodes u , v and edge ( u , v ) are added to G (unless they have been already added). The weight of ( u , v ) is set to be the negative logarithm of the Sulston score between clone c i and clone c j . The objective of the bipartite matching is to attempt to group together clone fragments that are located at the same location on the genome. Because of the noise in the fingerprint data, some of the matched fragments might not represent the same region in the target genome. In the following steps, we try to eliminate as many false matches as possible. MFG pruning. In this step, some of the components of G that might represent more than one unique region in the target genome are split. Specifically, we examine all the connected components of G and mark the ones that satisfy at least one of the following conditions as candidates . 1. Extra fragment: The connected component contains multiple fragments of a clone. 2. Unmatched fragments: The di ff erence between the length of the second shortest and the second longest fragment in the connected component is more than the tolerance value T . We ignore the shortest and the longest fragments to allow two outliers per component. 3. Weak overlap: The connected component contains at least one pair of clones that are very unlikely to overlap (i.e., have Sulston score of at least 1e-2.5 for MTP-ILP or 1e-1 for MTP-MST). For each candidate component, a min-cut algorithm is used to partition it by removing the weakest set of edges (i.e., minimum total edge weight) [11]. After a component is partitioned into two subgraphs, we check both subgraphs against the three conditions. If at least one is satisfied, we partition that component again. This iterative process terminates as soon as there are no more components that need to be split. Elimination of spurious components. In order to introduce the notion of spurious components, let us assume for the time being that the exact locations of the clones in a contig are known. Let U be a connected component of G , and let N ( U ) be the corresponding connected component cloneset. We call U spurious if the overlapping region for the clones in N ( U ) is spanned by another clone in the contig. An example is illustrated in the Appendix. This step is crucial to reduce the number of redundant clones in the preliminary MTP produced by MTP-ILP. Recall that MTP-ILP selects the smallest set of clones that cover each connected component in the MFG. If the spurious components are not removed, at least one clone in each spurious component is added to the MTP without contributing to the overall coverage. Since the exact ordering of the clones is not available, we employ a probabilistic method to detect spurious components. More specifically, we compute the probability that a connected component is observed purely by chance. If this probability is high then we mark the component as spurious. We expect the number of spurious components to be low, since missing fragments or extra fragments are unlikely to occur frequently for a specific cloneset. In our model, we assume that fragment sizes are distributed uniformly, and that the probability that two clones share a fragment is P = 2 T / gellen [19], where T is the tolerance and gellen is the number of possible fragment size values. The probability that two clones share f fragments is P f and the probability that c clones share f fragments is P ( c − 1) f . This suggests that the probability that a connected component U is observed Removing clones that are buried into multiple clones. Recall that the goal of the first step in the construction of the MFG is to bury clones into other clones to reduce the problem size. In this step, we reduce the problem size further by removing clones that are buried into multiple clones. For example, in Figure 3 in the Appendix, clone C is buried into B ∪ D , thus C can be removed. For each clone in a contig, we compute the ratio between the number of its fragments in the MFG and the total number of its fragments. If at least B % of its fragments are already in the MFG, then we mark that clone buried. However, there is a complication. It is possible that two clones are mutually buried and therefore we cannot remove both. In order to solve this problem, we first identify all possible candidates that can be buried into multiple clones. Then, we examine each connected component U of G , and save N ( U ) in a list L if all clones in N ( U ) are candidates to be buried. All candidates that do not exist in any N ( U ) ∈ L can be buried. However, we need to make sure that at least one clone in each N ( U ) ∈ L is not buried, otherwise the region in the target genome that is represented by U is not covered. So the task at hand is to remove as many candidate clones as possible with the constraint that at least one clone survives from each cloneset N ( U ) ∈ L . This problem is called minimum hitting set and it is NP-complete (polynomial reduction from the vertex cover problem [8]). Here, we solve it sub-optimally using a greedy approach [8]. At each iteration, we (1) select the clone c that occurs in the maximum number of clone sets in L , (2) remove all clone sets that contain c , (3) save c into minimum hitting set H , and (4) repeat. The iterative process terminates when the list L is empty. Buried clones selected in this way are removed from the MFG permanently. If this removal introduces components with only one node, they are also removed from the MFG permanently. The algorithm is summarized as Algorithm 1 in the Appendix. in Removing Adding the MFG mandatory clones is P that clones , are where buried to the f is MFG. into number multiple Clones of times clones. that that have Recall N to ( U be ) forms selected that the a connected goal as MTP of the clones com- first ponent. step are called in the In mandatory our construction algorithm, clones. of if the ( | U More MFG | − 1) specifically, f is is to lower bury than clones if at a least threshold into 50% other Q of clones then the fragments we to mark reduce U of the as a spurious. problem clone are size. Spurious not In present this components step, in the we MFG, reduce are removed then the problem that from clone the size is MFG further marked permanently. by as removing mandatory. clones When that a are clone buried is marked into multiple as mandatory, clones. it For is example, immediately in Figure 3 stored in in the the MTP Appendix, and ignored clone C for is buried further into analysis B ∪ D by , thus removing C can be all removed. of its fragments from the MFG. For each clone in a contig, we compute the ratio between the number of its fragments in the MFG and the total number of its fragments. If at least B % of its fragments are already in the MFG, then we mark that clone buried. However, there is a complication. It is possible that two clones are mutually buried and therefore we cannot remove both. In order to solve this problem, we first identify all possible candidates that can be buried into multiple clones. Then, we examine each connected component U of G , and save N ( U ) in a list L if all clones in N ( U ) are candidates to be buried. All candidates that do not exist in any N ( U ) ∈ L can be buried. However, we need to make sure that at least one clone in each N ( U ) ∈ L is not buried, otherwise the region in the target genome that is represented by U is not covered. So the task at hand is to remove as many candidate clones as possible with the constraint that at least one clone survives from each cloneset N ( U ) ∈ L . This problem is called minimum hitting set and it is NP-complete (polynomial reduction from the vertex cover problem [8]). Here, we solve it sub-optimally using a greedy approach [8]. At each iteration, we (1) select the clone c that occurs in the maximum number of clone sets in L , (2) remove all clone sets that contain c , (3) save c into minimum hitting set H , and (4) repeat. The iterative process terminates when the list L is empty. Buried clones selected in this way are removed from the MFG permanently. If this removal introduces components with only one node, they are also removed from the MFG permanently. The algorithm is summarized as Algorithm 1 in the Appendix. Adding mandatory clones to the MFG. Clones that have to be selected as MTP clones are called mandatory clones. More specifically, if at least 50% of the fragments of a clone are not present in the MFG, then that clone is marked as mandatory. When a clone is marked as mandatory, it is immediately stored in the MTP and ignored for further analysis by removing all of its fragments from the MFG. Recall that the MTP can be computed by selecting the smallest set of clones that cover all connected components of the MFG. This problem is a special case ...

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... The contig-wise coverage is computed for each contig and then an overall score is computed as the weighted average of contig- wise coverage, where the weight is the number of MTP clones in each contig. Although contig-wise coverage appears very similar to global coverage, it may produce di ff erent results when a region in the genome is covered by multiple contigs. Both FPC and FMTP have six parameters. Depending on the fingerprinting method (i.e., agarose or High Information Content Fingerprinting (HICF)), FMTP provides default values for its parameters. Using values for these parameters close to the defaults is crucial to obtain good performance. For example, the cuto ff parameter C should be changed slightly. By default, MTP-ILP uses a low C value (1e-10 for agarose or 1e-40 for HICF). Since MTP-ILP processes the original contigs which usually contain many clones, a higher value of C would introduce many false positive overlaps. On the other hand, to detect shorter overlaps, MTP-MST uses a high C value (1e-2 for agarose or 1e-10 for HICF). We have generated a large number of MTPs using both tools with several parameter sets, however we only recorded the best possible MTP for a given size (i.e., number of clones). If we obtained two MTPs M i and M j using di ff erent parameter values, if the size of M i is greater than the size of M j then the coverage of M i must be greater than coverage of M j , or otherwise M i is disregarded. As a result, in the experimental results as the size of the MTP increases, the coverage increases monotonically. In order to have a fair comparison between FMTP and FPC, we used the same B and T values used by FPC when building the physical map ( B = 90, T = 7 for rice, and T = 3 for barley). We set the C for the MTP-MST to values between 9e-2 and 5e-4. For all other parameters, we used the default values ( Q = 3, B = 80, C = 1e-2.5 for MTP-ILP, 1e-1 for MTP-MST, C = 1e-10 for MTP-ILP). The graphs summarizing the results are shown in Figure 2. Each point in the graphs represents an MTP. Figure 2-LEFT shows the contig-wise coverage as a function of the number of MTP clones; Figure 2-RIGHT illustrates optimal, global, and random coverage of the MTPs as a function of the MTP size. First, observe that when all clones in the rice contigs are selected as MTP clones, the global coverage is only 94.45% of the genome. As shown in Figure 2-LEFT and RIGHT, FMTP produces MTPs with significantly better contig-wise and global coverage than FPC, sometimes even with fewer clones. For instance, the highest possible contig-wise coverage that we were able to obtain by using FPC is 84.96%, whereas FMTP’s is 85.11% with about 460 (12%) less clones. This would imply 12% reduction in the sequencing costs. Also, global coverage of MTPs produced by FMTP converges to the optimum coverage much faster than FPC. The number of redundant clones and gaps produced by FPC and FMTP are almost identical and very small. The average overlap size between consecutive clones is smaller in FMTP than FPC (data not shown). This explains the di ff erence in coverages in Figure 2. Another interesting observation can be made by comparing the optimal coverage in Figure 2-RIGHT. The optimal MTP for FMTP has higher coverage and has a smaller number of clones than the optimal MTP of FPC. Recall that the optimal coverage is computed by selecting a set of clones for each contig that covers the widest possible region. A higher coverage (even when the number of clones is smaller) suggests that FMTP selects relatively more MTP clones from the “big” contigs than FPC. We ran FMTP and FPC on the barley physical map generated by our group at the University of California, Riverside and several other institutions [14]. FPC generated MTPs that contain between 11,000 and 21,000 clones. When default values are used, FMTP generated MTPs that contain about 18,000 clones. In terms of running time, FMTP and FPC are comparable. Both tools compute MTP in a couple of hours. We presented a set of novel algorithms to compute the MTP of a physical map by using a two-step approach. In the first step, we used a stringent threshold to reduce the problem size by generating a preliminary MTP without compromising the coverage of the contigs. In the second step, we attempted to order the clones in the preliminary MTP by computing MST of an overlap graph. Then, we ran a shortest path algorithm to compute the MTP. Our experimental results show that our method generates MTPs with significantly higher coverage than the most commonly used software FPC, even using a smaller number of MTP clones. Our experimental results also show that FMTP could reduce substantially the cost of clone-by-clone sequencing projects. The authors would like to thank Prof. Carol Soderlund, Dr. William Nelson, and the anonymous reviewers for insightful comments that helped improving the manuscript. This project was supported in part by NSF CAREER IIS-0447773 and NSF DBI- 0321756. In Figure 3, an illustration of five clones in a contig are shown based on their real coordinates in the genome. Suppose now that the MFG of this contig contains a connected component that contains fragments only from clones B and D . That component is spurious (i.e., does not represent a region in the target genome) because the overlap between clones B and D is completely covered by clone C . Spurious components arise for two reasons, namely missing fragments (in clone C in the example) or extra fragments (in clone B or D in the example). A sketch of the algorithm that removes clones that are buried into multiple clones is presented as Algorithm 1. A sketch of the algorithm that detects clone overlap is presented as Algorithm 2. At each iteration four conditions are checked to determine if u i and u i + d are overlapping where u i , 1 ≤ i ≤ | | , is the ith clone in . All conditions have to be true to add the edge ( u , u ) to . In line 7 and 8 of Algorithm 2 we check whether the clone pairs u i u i + d − 1 and u i + 1 , u i + d are overlapping. Obviously, if at least one of these pairs do not overlap then u i and u i + d cannot be overlapping (assuming that no clone is completely contained in another clone). In line 9, we check if clones u i , u i + 1 , . . . , u i + d have fragments together in at least one connected component of G . If u i and u i + d are overlapping then u i , u i + 1 , . . . , u i + d have to be overlapping, and therefore they should share at least one fragment in G . At the end, we check if S ( u , u ) ≤ C ...