text stringlengths 128 2.05k |
|---|
To better understand the behavior of the mean completion time, we illustrate in Fig. the effects that changes in the parameters [MATH] have on these mean completions times for the process in which the forward rate on the correct branch is six times the rate on the wrong branch, [MATH] At first glance at Fig. A or Fig. ... |
III.3 Variance in Completion Times In addition to specificity and the average time to arrive at that specificity, a completion process is further characterized by the shape of the distribution for its completion time. For some parameters this process will have little variance, and the decision is made in some seemingly... |
In what follows we consider the same cases as above and classify the shapes of the resulting completion time distributions. First, we consider the case of zero proofreading rates, [MATH] Fig. shows a contour plot of the coefficient of variation of the arbitrary completion time versus [MATH] and |
[MATH] and typical completion time distributions for the parameter values [MATH] and [MATH] [MATH] This plot allows us to divide the parameters space into a few regions with different shapes for the completion time distribution. The large green area (color online) in the upper right corner corresponds to [MATH] where t... |
[MATH] . In this case, the completion time distribution for each branch is a Gamma distribution, and the total completion time distribution is a simple combination of the two, since the probability to complete at each of the branches is proportional to the forward rate at that branch. As a result the total completion t... |
We now consider the case where there is proofreading ( [MATH] ) but where the backward rates are set to zero, [MATH] . Fig. shows a contour plot of the coefficient of variation of the arbitrary completion time versus |
[MATH] and [MATH] and typical completion time distributions for the parameter values [MATH] and [MATH] [MATH] As above in Fig. , we can divide the parameters space into few regions with different shapes for the completion time distribution. For example the large green area (color online) corresponds to a coefficient of... |
We now turn to the more general case where there is both proofreading and a backward reactions [MATH] [MATH] ). For this case, Fig. shows a 3D plot of the coefficient of variation of the arbitrary completion time vs. |
[MATH] (upper line) or [MATH] lower line. These figures emphasize the different effect of changes in [MATH] or [MATH] . While in all cases strong backward bias on both branches (large [MATH] or [MATH] lead to an exponential distribution of the completion time, backward bias has different dependence on the system size a... |
III.4 Simplification of the Two-Branch Decision Process In examining the distributions in Figs. A-D, one observes that the completion time distribution of each branch is often similar to a gamma distribution (or an exponential distribution, which is a special case of the gamma distribution). This suggests that one shou... |
[EQUATION] where [MATH] denotes the probability of completion in the first direction. Thus, the total probability density of completing along either branch at time [MATH] is approximated by: |
[EQUATION] In numerical studies, we have attempted to find parameter sets [MATH] that best match the direction and time distribution of the full escape process in the one norm sense. In other words, we have found the [MATH] such that: |
[EQUATION] In most cases, we find that this approximation and optimization does an excellent job of capturing the qualitative and quantitative behaviors of the complete process as is shown in Figs. A-D. To further explore the ability of the reduced model to capture the behavior of the full system, we have explored the ... |
As was the case for the AM process ( [MATH] ), the dKPR process ( [MATH] ) is well captured by the same three state process defined above. To illustrate this, the colored lines in Figs. A-D correspond to the full system completion time distributions, and the markers correspond to the approximate three state system. |
IV Conclusions In this work we have begun the exploration of the temporal properties of kinetic proofreading schemes. To accomplish this, we have derived analytical expressions for the Laplace transform of the occupation probabilities from which we obtained the completion time distributions. With this analysis, we have... |
We have explicitly considered different kinetic schemes including the traditional directed kinetic proofreading (dKPR) scheme where catastrophic reactions force the process to restart as well as an absorption mode (AM) where single step intermediate reactions can provide the same specificity. Surprisingly, we find that... |
High specificity appears in many biological systems and likely results from many different kinetic schemes–suggesting that one needs as much information as possible to distinguish between one such mechanism and the next. Therefore, in addition to using the specificity and mean completion times to compare the different ... |
Acknowledgements. We thank N. Hengartner for discussions during early stages of this work. We also thank B. Goldstein, R. Gutenkunst, M. Monine, and M. Savageau for helpful comments regarding this work. This work was partially funded by LANL LDRD program. |
# Source: arxiv 0909.3129 # Title: Copy-number-variation and copy-number-alteration region detection by cumulative plots # Sections: all # Downloaded: 2026-03-02T08:42:29.304697+00:00 |
Copy-Number-Variation and Copy-Number-Alteration Region Detection by Cumulative Plots (preprint version of BMC Bioinformatics , 10(suppl 1):S67 (2009) ) |
Abstract Background: Regions with copy number variations (in germline cells) or copy number alteration (in somatic cells) are of great interest for human disease gene mapping and cancer studies. They represent a new type of mutation and are larger-scaled than the single nucleotide polymorphisms. Using genotyping microa... |
Results: We apply the cumulative plot to the detection of regions with copy number variation/alteration, on samples taken from a chronic lymphocytic leukemia patient. Two sets of whole-genome genotyping of 317k single nucleotide polymorphisms, one from the normal cell and another from the cancer cell, are analyzed. We ... |
Conclusions: As a graphic tool, the cumulative plot is an intuitive and a scale-free (window-less) way for detecting copy number variation/alteration regions, especially when such regions are small. |
Background Most efforts in genetic mapping of human diseases focus on single-nucleotide-polymorphism (SNP): individual nucleotide base that may differ from one person to another. If the cause of a polymorphism is due to diverging paths in population genetic history, such as in multiple ethnic groups, it can be used as ... |
. If the polymorphism is a functional mutation (non-synonymous or promoter-region polymorphism) underlying a human disease, then it is the focus of attention in case-control genetic analyses |
A new type of genetic polymorphism emerged recently as another source of mutation that may lead to human diseases: the copy number variation (CNV) (for literature on CNV, see an online bibliography |
). Local duplication and deletion events occuring at kb ( [MATH] bases) or Mb ( [MATH] bases) scales are the cause of CNV. If these events occurred in prior generations, CNV can be treated as a genetic marker whose transmission might be traced in studying the disease-status correlation. These events can also occur in t... |
Similar duplication and deletion events also occur in somatic cells, leading to copy number alteration (CNA). Besides the link between CNA and cancers studied before |
, an early CNV-disease association was reported on Charcot-Marie-Tooth disease , in inherited neurological disorder. In the past year or two, the number of reports on association of CNV with human diseases increased dramatically, especially for psychiatric disorders such as Schizophrenia |
bipolar , and for brain developmental disorder such as Autism . These diseases have long been evading genetic dissection, and the CNV link offers new optimism for our ultimate understanding of these diseases. |
The technology for CNV detection evolves from Mb-level comparative genomic hybridization (CGH) to higher-resolution array-based CGH |
Genotyping array whose original goal is to genotype individual SNPs, has increasingly been used for CNV detection There are two relevant pieces of information from a genotyping array data for the purpose of CNV detection. |
The first is the ratio of intensity reading of alleles for a sample to that from a reference group of normal samples. If the ratio is larger than 1, there are more copies of piece of DNA in the sample than normal (which is 2 copies). If the ratio is less than 1, it indicates a deletion. The second signal is the genotyp... |
The homozygosity property of one-copy deletion is well exploited in detecting loss-of-heterozygosity in CNA of cancer cells CNV detection using genotype microarray data relies on these two sequences: if the intensity ratio deviates from the normal value of 1 for a chromosome region with a consistent value, it can be a ... |
Methods for calling CNV regions can be roughly classified into two types. The first type is straightforward: a CNV detection is claimed when the log-ratio value is significantly deviated from 0 |
. The problem with this method is that the threshold for calling CNV varies greatly from platform to platform, from study to study, and a comparative investigation is urgently needed |
. The second type uses hidden Markov models (HMM), where the underlying CNV status is the hidden variable, and the log-ratio and genotype sequences are the two observed variables |
One advantage of the HMM framework is that it can incorporate information from both sequences at once. When the parameter settings in a HMM are fixed, HMM is a stationary (homogeneous) process along a chromosome. There is one parameter in HMM which controls the transition probability from the (hidden) CNV state to non-... |
. What if the CNV regions do not have a characteristic size, or equivalently, the length distribution is not exponential? In that case, CNV-calling methods that do not require stationarity are preferred. |
The guanine-cytosine content (GC%) in DNA sequences has been a focus of non-stationary, non-Markov, long-range-correlated modeling for more than twenty years |
It is well acknowledged that the hierarchical pattern of GC%-domains within GC%-domains is possible In order to detect both small and large GC-homogeneous domains, one applies methods that do not preset a characteristic scale. One such method is the recursive segmentation that adopts a divide-and-conquer approach |
. Another is the cumulative plot. Cumulative plot is a graphic display of sequence information such that trend in a region becomes more visible and obvious. It is a window-less method because no characteristic scale needs to be specified, although a window can be imposed to a plot when all patterns within certain lengt... |
or “Z curve” The cumulative plot has also been widely used for detection of replication origin . To our knowledge, cumulative plots have not been applied to CNV/CNA detection. The purpose of this paper is not to provide a comprehensive comparison of various CNV/CNA-calling methods, but limited to the presentation and i... |
Results and discussion Since our method applies equally to CNV and CNA data, here we examine the CNA pattern in a cancer patient with chronic lymphocytic leukemia (CLL) |
. DNA samples from the patient’s normal cell and that from the cancer cell are obtained and genotyped with 317,000 SNPs genomewide. Figure 1 shows the log-ratio and [MATH] sequences (see Methods) for chromosome 13, where a 9Mb CNA region (deletion) in the cancer cell is clearly visible. A deletion region is characteriz... |
The left panel of Figure 2 shows the two cumulative plots corresponding to log-ratio sequence and homozygosity indicator sequence [MATH] , respectively. In the simplest version, at each new SNP, the curve moves up or down by an amount equal to the log-ratio value of that SNP, or by the presence of a homozygote (+1) and... |
For a deletion region, the log-ratio value is consistently negative, and the first cumulative plot shows a drop; and genotype is consistently homozygous (also called run of homozygosity (ROH)), and the second cumulative plot shows a jump. However, from Figure 2 (left), even outside the CNA region, the first (second) cu... |
To remove the global or chromosome-wide average, we redraw a detrended cumulative plot (right panel of Figure 2) where the linear trend from the normal cell is subtracted from the two cumulative plots. If the difference of global trends between the cancer and normal cell is an artifact, e.g., the poor DNA quality in ca... |
Cumulative plots can be customized to pick any specially defined signal. Suppose we are mainly interested in regions with copy number equal to 1, i.e., hemizygous deletion. Such deletion region should exhibit two features: (1) log-ratio is equal to [MATH] 0.693147 (as versus [MATH] in the normal situation); (2) homozyg... |
Figure 3 shows the cumulative plot for “one deletion” indicator variable, without or with detrending (by the linear trend in the normal sample). In both versions, the hemizygous deletion region can be seen clearly. Not only the cumulative plot detects the CNA region easily, but also it delineates the border of the dele... |
When deletion occurs in both chromosomes, called homozygous deletion, the copy number is equal to zero. For homozygous deletions, both A- and B-channel intensity (see Methods) is close to zero, and the log( [MATH] ) is a large negative value. Because in the A- and B-channel plane (see Methods), these SNPs are near the ... |
We define a “two deletions” indicator variable whose value is 1 if the log-ratio is [MATH] ; and the value is [MATH] otherwise. Note that the genotype information is not used. Figure 4 shows the cumulative plot for the “two deletions” indicator variable for chromosome 13. One homozygous deletion region with [MATH] 1Mb ... |
The 9Mb deletion on chromosome 13 in our CLL sample, which was one of the known common deletions for this disease , represents an example of easy detection of CNA/CNV region, because the difference between the normal and cancer cell for both log-ratio and genotype sequence is already obvious from the raw data (Figure 1... |
Figure 5 shows the example of chromosome 6 of our sample where there is no large-scaled CNA region. The log-ratio and genotype sequence look almost identical between the normal and the cancer cell. The cumulative plot for the “one deletion” indicator variable shows that there are +400 more SNPs in the cancer cell than ... |
In order to explore the possible existence of smaller CNA regions, we pick the longest ROH region (roughly 4Mb) and view it with cumulative plots. Figure 6 (left) shows the un-detrended cumulative plot for the one-deletion indicator variable in this region. A clear hemizygous deletion region should show up as a jump in... |
The failure in detecting hemizygous deletion at the Mb scale does not necessarily prevent its possible existence at a smaller length scale. The right panel of Figure 6 shows a 200kb sub-region (marked in Figure 6 (left)) that contains a 36kb region with an upward trend in the cumulative plot. A zooming into any small r... |
It was previously suggested that run of homozygosity can be a sequence feature that is associated with certain human diseases We see here that ROH is only a partial indicator for a CNA/CNV region. The longest ROH on chromosome 6 in our sample only shows some weak evidence in a much narrower region for one-deletion CNA.... |
The pairing of the normal and the cancer sample is not essential to our method. In Figures 3,4,6, the CNA regions can be identified by cancer sample (the blue curve) alone. However, the comparison with the normal sample provides supporting evidence that deletion only occurs in the cancer cell and not in the normal cell... |
When SNPs along a chromosome are not evenly distributed, it may not be appropriate to move one step per SNP in the cumulative plot. For example, if multiple SNPs are in strong linkage disequilibrium in a densely typed region, the indicator variable values are positively correlated, and a sequence of +1 values is partia... |
(see Method) in favor for concordant genotypes between neighboring SNPs, as compared to the average. If [MATH] , we discount a +1 (or [MATH] ) movement by dividing the [MATH] value. For the chromosome 13 data, [MATH] is in a very narrow range of (0.9921, 1.0002). Because the probability ratio in favor of concordant hom... |
So far the delineation of an upward trend in the cumulative plot is determined by visual inspection. Segmentation programs can be developed to carry out the delineation automatically. In particular, one may move along the cumulative plot, calculate the slope from the start point to the moving position, then from the mo... |
Finally, for case-control analysis using CNV, one deals with two groups of samples . In this situation, cumulative plot can be first applied to each individual person to identify the CNV/CNA region. Then, chromosomes can be partitioned into equal-sized windows and the frequency of CNV/CNA-containing window in the case ... |
Conclusions We have shown here that cumulative plots of an indicator variable derived from the log-ratio and SNP genotype sequence can easily identify CNV or CNA regions. We illustrate the procedure for hemizygous deletion (copy number equal to 1) and homozygous deletion (copy number equal to 0) using samples taken fro... |
Methods log-ratio and genotype data In a two-channel (two-color) SNP genotyping microarray, the A- and B- channel (A- and B-allele) intensity reading is recorded. These two intensities are normalized by reference intensity values which are obtained by averaging many normal samples. Each SNP can be represented by a poin... |
and [MATH] . Log(r) is the “log ratio” value that provides a copy-number information, and [MATH] provides a genotype information, where [MATH] |
correspond to two homozygotes, and [MATH] corresponds to the heterozygote. Note: (1) [MATH] value depends on a group-averaged reference level, and this information is provided by the array-maker company. (2) Although [MATH] |
and [MATH] is in principle independent, there could be weak correlation between them. Our starting point are the two sequences of log(r) and discretized [MATH] values (i.e. genotype) along a chromosome. |
Cumulative plots for log-ratio and homozygosity sequence The [MATH] and [MATH] variable is transformed by: log-ratio [MATH] and homozygosity indicator [MATH] For heterozygotes, [MATH] is close to [MATH] , and for two homozygotes, |
[MATH] is close to 1. Denote the [MATH] -th SNP’s log-ratio and homozygosity indicator as [MATH] and [MATH] . The (original) cumulative plots of these two sequences are: |
[EQUATION] A cumulative plot can be detrended such that the first and the last SNP are on the same horizontal line. The purpose of this detrending is to remove the chromosome-wide bias so that regional deviations are highlighted. In our normal and cancer cell from the same individual example, we detrend the normal samp... |
[EQUATION] To highlight the difference between the cancer cell and the normal cell, we use the [MATH] and [MATH] obtained from the normal cell to detrend the cumulative plot for the cancer cell. |
Cumulative plots corrected by spacing between neighboring SNPs When SNPs are not distributed evenly along a chromosome, one may consider correcting the effect of inhomogeneous correlation between neighboring SNPs. We first calculate the probability of a neighboring SNP of a homozygous SNP to be also homozygous due to t... |
Haldane’s map relates the number of recombinations within a chromosomal interval [MATH] and the probability of observing a recombinant between the two end points [MATH] |
[EQUATION] The unit of [MATH] is Morgan, which is roughly equal to 100Mb (or 1 centi Morgan is equal to 1 Mb ). The probability of observing a non-recombinant is [MATH] |
Denote [MATH] the probability that one homozygous SNP is followed by another homozygous SNP that is [MATH] genetic distance apart. Since Haldane formula is applicable to haplotype, or a single copy of a chromosome, for two copies of a chromosome, |
[MATH] Suppose the average spacing between two neighboring SNPs is [MATH] For a neighboring SNP pair whose spacing [MATH] it is more likely for both SNPs to be homozygous than the average, by a probability ratio of [MATH] and the cumulative plot for the homozygosity indicator variable can be adjusted by dividing that r... |
[EQUATION] We assume that [MATH] is calculated in the same way as for other indicator variables, meaning CNV/CNA of a particular type is maintained at the neighboring SNP by the same probability [MATH] , and the above formula can be used to correct other cumulative plots. Note that transition probability from one genot... |
Authors contributions W.L. designed the method, carried out the analysis, and wrote the manuscript; A.L. genotyped the samples; P.K.G. proposed the CNV study of chronic lymphocytic leukemia. |
List of abbreviations CGH: comparative genomic hybridization CLL: chronic lymphocytic leukemia CNA: copy number alterations CNV: copy number variations |
GC%: guanine and cytosine contents HMM: hidden Markov models ROH: run of homozygosity SNP: single nucleotide polymorphism Acknowledgements |
We thank Nick Chiorazzi for providing the CLL sample, and Pedro Bernaola-Galván, José Oliver for discussions on segmentation methods. |
Figures Figure 1 - Log-ratio and genotype sequences for chromosome 13 in paired samples from a CLL patient Log-ratio (top) and genotype [MATH] (bottom) sequence for SNPs from chromosome 13 of two samples taken from the same cancer patient: black for normal cell and blue for cancer cell. For the log-ratio plot, the copy... |
[MATH] are marked. Figure 2 - Cumulative plot of log-ratio and homozygosity sequence Cumulative plot and detrended cumulative plot for both the log-ratio sequence and the homozygosity indicator sequence (for the chromosome 13 data shown in Figure 1). Top: cumulative plots for log-ratio sequence. Bottom: cumulative plot... |
[MATH] for heterozygote). Left: original cumulative plots. Right: detrended cumulative plots. The linear trend obtained from the normal sample is used to detrend both the normal and the cancer sample. Black for the normal cell and blue for the cancer cell. The 9MB hemizygous deletion and the neighboring 1Mb homozygous ... |
Figure 3 - Cumulative plot of the hemizygous deletion indicator variable Cumulative plot (top) and detrended cumulative plot (bottom) for the 9Mb hemizygous deletion region on chromosome 13, using the “one deletion” indicator variable. |
Figure 4 - Cumulative plot of the homozygous deletion indicator variable Cumulative plot (top) and detrended cumulative plot (bottom) for the 1Mb homozygous deletion region on chromosome 13, using the “two deletions” indicator variable. |
Figure 5 - Log-ratio and genotype sequences for chromosome 6 in paired samples from a CLL patient The log-ratio sequence (top), genotype [MATH] sequence (bottom), and the detrended cumulative plot for the “one deletion” indicator variable for SNPs on chromosome 6. Black and blue color refer to the normal and cancer cel... |
[MATH] are marked in the log-ratio plot. Figure 6 - Zoom in of smaller regions Cumulative plots of “one deletion” indicator variable for the region marked in Figure 5 (left), and the sub-region marked by a horizontal bar on the left (right). Black and blue refer to the normal and the cancer sample. |
# Source: arxiv 0909.3384 # Title: Comparing Single and Multiobjective Evolutionary Approaches to the Inventory and Transportation Problem # Sections: all # Downloaded: 2026-03-03T01:57:33.995371+00:00 |
Comparing Single and Multiobjective Evolutionary Approaches to the Inventory and Transportation Problem (August 2009 Draft submitted to Evolutionary Computation |
Abstract EVITA, standing for Ev olutionary nventory and ransportation l-gorithm, is a two-level methodology designed to address the Inventory and Transportation Problem (ITP) in retail chains. The top level uses an evolutionary algorithm to obtain delivery patterns for each shop on a weekly basis so as to minimise the ... |
The aim of this paper is to investigate whether a multiobjective approach to this problem can yield any advantage over the previously used single objective approach. The analysis performed allows us to conclude that this is not the case and that the single objective approach is in general preferable for the ITP in the ... |
Introduction Given a retail chain and a central depot that supplies it, both belonging to the same company, we define the Inventory and Transportation Problem as that whose objective is to minimise the costs of both inventory and transportation, subject to a number of constraints imposed at the shop level. |
In previous work Esparcia-Alcázar et al., 2006a, Esparcia-Alcázar et al., 2006b, Esparcia-Alcázar et al., 2007a, Esparcia-Alcázar et al., 2007b, ; Esparcia-Alcázar et al.,, 2009 we employed a single objective approach to this problem, which aimed at minimising the total weekly cost calculated as the sum of the inventor... |
With this aim in mind we have carried out an extensive series of experiments using the eight instances used in (Esparcia-Alcázar et al.,, 2009 , plus two new ones . The set of restrictions, consisting of the characteristics of the vehicles employed and the working hours of the drivers (see Table |
), plus the parameter configuration of the evolutionary algorithm are kept as in that work. In the multiobjective approach we have used the NSGA-II algorithm |
(Deb et al.,, 2000 , which is described in more detail in Appendix I. A further objective is related to the fact that the choice of the algorithm that yields the transportation routes (the VRP solver ) can play a significant part in the performance of the whole algorithm Esparcia-Alcázar et al., 2006a, . Here we will e... |
(Cordeau et al.,, 1997 , ant colony optimisation (Dong and Xiang,, 2006 and a classical VRP solving technique, Clarke and Wright’s algorithm |
(Clarke and Wright,, 1964 The rest of the paper is structured as follows. Section provides background on the problem; it contains a summary of the state of the art in this and related problems and a detailed description of the ITP. The top and lower levels of the algorithm are described in Sections and |
, the latter containing the details of the three VRP solvers employed. The experimental setup is given in Section , with results and analysis thereof contained in Section . Finally, Section |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.