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EMPSC A New Method Based on Ellipsoidal Model for Protein Structure Comparison

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EMPSC: A New Method Based on Ellipsoidal Model for Protein StructureComparisonYhi Shiau1,2, Jia-Nan Wang1, Yu-Feng Huang1, Chien-Kang Huang3yshiau@, jnwang@.tw, yfhuang@.tw, ckhuang@.tw1Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan2Chunghwa Telecom Laboratories, Tauyuan, Taiwan3Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, TaiwanAbstractThis paper proposes a new method EMPSC for the well-known PSC (Protein Structure Comparison) problem. The proposed method EMPSC is a protein structural alignment algorithm based on ellipsoidal model abstraction. We segment the protein 3D structure into two different kinds of structures, including Secondary Structure Elements recognized by DSSP1 and other coil/loop structures. These SSEs will be the initial alignment center for obtaining the transformation coordinate systems. Different heuristic filters and geometric hashing based global alignment estimation are used for quick finding better initial alignments. In the refined alignment stage of analysis, a standard refinement algorithm is invoked to fine-tune the alignment outputted by the first stage. Our experimental results reveal that EMPSC generally achieves comparable accuracy and better performance in comparison with the existing PSC algorithms. Moreover, we analyzed the factors that affect the EMPSC performance and SSE-based PSC algorithms. Further investigation in multiple protein structure comparison and local structure comparison will be continued.Keywords: Eigenvector, Dynamic Programming, Geometric Hashing, Secondary Structure Elements. Availability: The EMPSC tool is accessible at our lab website .tw IntroductionSince 1747 Beccari discovered proteins, the proteins play the important role in biochemical reactions, the study of protein functionalities attracted biochemical researchers. Within these research topics, Protein Structure Comparison (PSC) is one of the most basic and important subjects to detect the evolutionary and functional relationships between them. And we know that the functionality of one protein is related to its 3D structure2, that is, proteins with similar substructures may have similar functions. Therefore, improving the methodology and tools of PSC is an important issue in molecular biology and bioinformatics for many years3-7.Today the biochemists need more fast and accurate PSC tools, as the protein database grows fast withthe help of computation power in the recent biochemistry research. For examples, the Protein Data Bank8 content increases to 35,813 proteins on 28-Mar-2006, the Swiss-Prot content increases to 163,000 and the TrEMBL increases to 1,450,000. Obviously, continuing to improve PSC tools to handle the fast growing massive protein data is a great challenge.In order to detect the functional or evolutionary relationships between proteins, the PSC algorithms compare the 1D sequences information or 3D structures information of amino acid sequences. The purpose of PSC is to identify maxima equivalent Cα atoms upon which to optimally align the 3D structures of compared proteins. Previously proposed PSC algorithms exploit many different computing approaches, including Monte Carlo (Dali9), Dynamic programming10-12 (V AST13), 3D clustering14, graph theory15, spline approximation16 and geometric hashing17. In order to further speed up the PSC performance, quick-n-dirty approaches is applied in today’s PSC, like CE18, SAP19, ProSup20,21, and FLASH22. As the PSC is the NP-hard problem, most approaches tried to propose different heuristics to approximate the optimal solution. Therefore, quick-n-dirty approaches are the main stream of today’s PSC algorithms. In the following section, we will further explore these quick-n-dirty approaches.CE method started from the concept of aligned fragment pairs (AFPs), that is, the initial alignment finding is grown from AFPs. CE uses a combinatorial extension of an alignment path defined by AFPs to obtain the extension paths that pass the similarity threshold defined by the inter-residue distance constraint. The inter-residue distance constraint is an approximation for superimposing the two proteins. In refinement, the top-20 alignment paths are evaluated based on r.m.s.d. and the best one selected. The best alignment path is further refined with the dynamic programming approach to obtain the final global alignment solution. The problem of CE method is it is time consumed while calculating of inter-residue distance constraint. On the other hand, according to the observation of our experiments, we found CE method tends to find most corresponding residues rather than smaller r.m.s.dSAP method uses iterated double dynamic programming and applied it in both the initial alignment finding and the refinement processes. The initial alignment finding begins from every residue pair; each residue from the compared proteins. SAP finds out the potentially equivalent residue pairs based on local structure and environment. Then, for each potential equivalent residue pair, SAP calculates the alignment score with dynamic programming approach. The parameters of the scoring function include the direction component, the orientation component, the sequence term, and the spatial term. The initial alignment finding is done by sorting all the elements of the bias-matrix and taking a number (top-20) of the highest scoring elements. After initial alignment finding, top-20 results are further refined with the dynamic programming approach. In order to avoid the exhaustive search of all possible residue pairs, the useful option of taking a randomized selection has been exploited in SAP results. The main issue of SAP is the higher time complexity because double dynamic programming isvery time consumed for large matrix.ProSup method tried to approximate the solution from seed pairs. The seed fragments are the similar fragments of compared proteins. ProSup superimposes all possible seeds and evaluates the whole protein with the superimposing equivalences approach instead of using dynamic programming. In refinement, a standard procedure that combines dynamic programming and least-square constraint23,24 further refines the initial alignments. ProSup can output multiple solutions. The accuracy and efficiency is affected by the length of seed fragments.FLASH method greatly reduces the time complexity by with an aggressive abstraction of protein structures. FLASH identifies the Secondary Structure Elements (SSEs) of compared proteins with DSSP tool. It works on a vector-represented SSE as a data reduction for protein’s 3D structure. The experiments revealed that the SSE in the protein provides better initial alignment performance. FLASH only considered the information from α-helix and β-sheet, and neglected the coil information. It establishes the angle-distance map for all SSE pairs. After calculating the SSE matching probabilities, FLASH uses a greedy procedure to select viable alignment solutions (at least 3). Statistical significance is applied in FLASH to filter out inappropriate initial alignment findings. In refinement, FLASH applies the same standard refined alignment algorithm as ProSup. With SSE-based data reduction, FLASH greatly speedup the time complexity in initial alignment finding. And it can output multiple solutions. However, if two proteins have similar local structure, but the global similarity doesn’t exceed the criterion of statistical significance, then FLASH stops further comparison and won’t give any result. In addition, if the protein has no SSE structure, then FLASH method doesn’t apply. The single vector representation also raises problems. To represent α-helix in a single vector is proper, as the α-helix is hard to bend. But the representing vector of β-sheet may lose some structure information. As the β-sheet structure is usually bending or curved, the length of identified β-sheet will affect the derived single vector effectiveness seriously.Based on previous study, we hope to develop a new method which is more efficient than conventional PSC approaches without the limits of FLASH. In this paper, we propose a new protein structural alignment method based on ellipsoidal model, named as EMPSC, which applies generic heuristic mapping function in initial alignment stage. There are four steps in EMPSC method – preprocessing, initial alignment, refinement and final evaluation. EMPSC identify the SSEs of the target protein with DSSP tool. The remaining parts of this protein, mostly the coil/loop structure, will also be considered. Rather than using single vector representation as FLASH, we use ellipsoid model to represent each sequence segment. The detailed algorithm describes in the Proposed Method section.EMPSC has many better characteristics of previous approaches. The initial alignment finding is selected from segment pairs (mainly SSEs) instead of residue pairs. EMPSC can also output multiple solutions. Like FLASH, we believe that SSEs is clearly more important for protein’s structureconformation. And abstract the structure information with SSEs (α-helix, β-sheet and coil) could be better than CE’s AFP and ProSup’s seed. In addition, EMPSC like CE and ProSup, the initial alignment finding come from the view of local alignment.Proposed MethodThe workflow diagram of EMPSC is depicted in Fig 1, and the algorithm will be described in details. The algorithm basically has four steps.(1)Preprocessing: Segment the proteins into SSEs with DSSP tool, and generate theellipsoidal representation for each segment (mainly α-helix, β-sheet).(2)Initial alignment: Generate the potential aligned segment pairs, looking for a good initialalignment via heuristic filtering.(3)Refinement: Iteratively apply a dynamic programming algorithm to refine the initialalignment, which is a well-known procedures as most PSC algorithms.(4)Final evaluation: Evaluate the refined alignments and provide the number ofcorresponding residues and r.m.s.d of alignment solutions.Fig 1. The workflow diagram of the EMPSC algorithm.Preprocessing: Generate Ellipsoidal RepresentationAs most modern PSC algorithms, in the initial alignment step, we don’t estimate the alignment quality by comparing the structures of two proteins atom by atom, but comparing their abstract models. The ellipsoid model rather than all residues of SSEs, were used to represent the proteins’ overall 3D structures in EMPSC. The step (1) of the EMPSC algorithm is the process of generating ellipsoidal representation for each segment of the target protein. At first, in step (1a), we use the DSSP tool to identify the SSEs (α-helix, β-sheet) of the protein. In step (1b), contiguous residues of remaining will be clustered into one segment. Therefore, the remaining residues of this protein (that is the coil/loop information) are further clustered to a set of new segments (coil sub-segments) according the adjacencies of residues. After that, SSEs will be represented by a set of 3D-ellipsoidal model. Fig 2 depicted the relationship between residue chain and 3D-ellipsoidal model. The PCA (Principal Component Analysis)23 is applied in finding their 3 orthogonal eigenvectors and 3 respectiveeigenvalues. According to the steps (1c), we can decompose the protein into a set of residue segments and ellipsoidal representations as Fig 3.Residue chainFig 2. A 3D-ellipsoidal ball and their 3 eigenvectors(a) (b) (c)Fig 3. The generating process of ellipsoidal representation. The figure (a) displays the example protein. The figure (b) shows that the protein was processed by DSSP and the SSEs is identified. The figure (c) shows thefurther processing of remaining segments.Initial Alignment: Find a Good Superimposed Transformation for the Initial AlignmentsFinding good superimposed transformations are the main task in obtaining initial alignments of EMPSC. The whole process is described in step (2) of the EMPSC algorithm. In general, the initial alignment searching is a filtering process that eliminates impossible or dissimilar segments mapping and finds the top-N best segments pairs, that is, top-N superimposed transformations between two compared proteins.At first (2a), we generate all possible initial alignments from the compared proteins. In this step, every pair SSE segments, including α-helix and β-sheet only, of the compared proteins can be the center of the new coordinates, and the remaining coil sub-segments will be used in the biochemical filtering step (2b).Before estimating the quality of every initial alignment in step (2d), EMPSC will first check the localstructure of each initial alignment, that is, EMPSC will calculate the similarity between the two mapping SSE pair (α-helix and β-sheet) and the at most four surrounding coil sub-segments. In step (2b), EMPSC will further filter the initial alignments with a heuristic filtering function. Conceptually, we can define an integrated filtering function combining all factors that is effective for the judgment of good initial alignment. However, instead of implementing one integrated filtering function, we currently implemented several subsequent filtering processes which filter out unmatched or dissimilar pairs. These processes include three different kind of filtering – type filter, mass filter, and biochemical filter. The type filter makes sure the secondary structure types of mapping segments are the same (such as α-α, or β-β). The mass filter makes sure the difference of reside numbers between the two mapping segments are must less than four residues. The biochemical filter checks the similarity of biochemical properties between two segments. We found that the biochemical properties of coil segments that before or after the SSEs (α-helix and β-sheet) are beneficial for initial finding, rather than biochemical properties of SSE itself. Therefore, in this filtering process, EMPSC makes sure biochemical features of the surrounding coil sub-segments are similar. The biochemical feature is currently defined as the ratios of hydrophobic residues, polar with uncharged resides and polar with charged residues in compared segments. The detail of biochemical filter is described as following.The remaining candidates mapping SSE segments need to pass all filtering criteria. In step (2c), EMPSC aligns the geometric center and the 3 primary eigenvectors of the candidate mapping segments, and then, new coordinates for the two compared proteins will be generated, as Fig 4.In step (2d), a fast global alignment estimation based on geometric hashing is developed to estimate the quality of the initial alignments. The geometric hashing is a fast way to compare the 3D structure. Take the geometric center of mapping SSEs as the origin, the position of each Cα atom of one target protein is transforms to the polar coordinate system. In the preprocessing of geometric hashing, Every Cα atom will be put into the hash table according to the distance between the Cα atom and the origin. We also transform the other protein into new coordinate, and calculate the estimated alignment score. The estimated scoring function of global alignment is described in Fig 5. Finally, only top-N candidates will be selected for further refinement. That is, we reserve the top-N superimposedtransformations as the good initial alignments.Fig 4. Align the 3 eigenvectors of two ellipsoids according to their magnitude.Fig 5. The global alignment estimation based on geometric hashingRefinement and Final EvaluationThe step (3) applies the same refined methods as most quick-n-dirty PSC algorithms. The least square method is applied in refinement process, and the step (3) will repeatedly refine the initial alignment solutions until the number of corresponding residues converges. In this step, the algorithm will iteratively tune the global alignment center with superimpose two proteins using the dynamic programming approach. Finally, in step 4, EMPSC will output the tuned alignments of the top-N candidate from step (2) as the N alternative solutions.Complexity AnalysisIn data preprocessing, EMPSC will clusters the protein to form a set of ellipsoids with the DSSP tool and proposed ellipsoid clustering method. The ellipsoid clustering is very fast (less than 1 second),and the time complexity is O(r) where r is the number of residues of the segments. In the initial alignment finding stage, the time complexity for EMPSC algorithm in this stage is O(e log e + pn) withthe scoring function based on fast the O(n)hash function, where n is the number of residues in the protein, e is the number of segments in the protein, p is the number of mapping SSE candidate segment pairs and p is much smaller than e. The complexity of refined alignment stage is O(Cn2), while the C is number of iterations before the refinement process is converged for each initial alignment. According to the observation, the number of iterations in EMPSC is usually less than 10.In the discussion section, we will find how the refinement process affects the execution time of EMPSC.Experiments and ResultsThree experiments are tested in order to test EMPSC in different conditions of protein structure comparison problems. In each experiments, EMPSC provides maximal 10 alternative solutions, that is,the Top-10 initial alignments in EMPSC are selected. These results reveal the efficiency and effectiveness of EMPSC in comparison with Dali, CE, V AST, ProSup, and FLASH. The results ofDali and ProSup are coming from the original papers. The results of CE and FLASH are gathered from our experiment environment, and they are consistent with the original papers. The computing environment for the experiments consists of Dual Pentium-4 Xeon 3.06GHz CPU and 2GB DRAM memory. All testing programs are not parallelized.One-against-all search for structural neighborsAs previous research work, we choose cAMP-dependent protein kinase to experiment onone-against-all search for structural neighbors. In order to compare with the existing results of Dali, CE, V AST, ProSup, and FLASH, the parameter d c (the distance cutoff for alignment construction) is assigned to 6Å. For all method, we listed the number of maximal correspondent residues and minimalr.m.s.d. Since we can find the program of CE and FLASH, we also list the execution time of CE and FLASH running in our computing environment. Comparing the value of r.m.s.d and number of corresponding residues, EMPSC can perform as well as other previous methods.Table 1. An experimental set of structural neighbors of cAMP-dependent protein kinase(1atp:E) identified bydifferent PSC methodsA sample set of structural neighbors of cAMP-dependent protein kinase (1atp:E)(336)ProSupFLASH EMPSC Dali CE V ASTProtein(residues) rmsd/#res rmsd/#res/sec rmsd/#res rmsd/#res rmsd/#res/sec rmsd/#res/sec 2cpk:E(336) 0.4 / 336 0.37/336/4.9 0.4 / 334 0.4 / 336 0.37/336/0.38 0.37/336/6.11apm:E(341) 0.3 / 336 0.33/336/4.94 0.3 / 334 0.3 / 336 0.33/336/0.51 0.32/336/6.191cdk:A(343) 0.4 / 336 0.38/336/4.95 0.4 / 334 0.4 / 336 0.38/336/0.39 0.38/336/6.36 1ydt:E(334) 0.5 / 334 0.45/336/4.5 0.5 / 334 0.5 / 334 0.45/336/0.31 0.45/334/6.03 1bkx:A(337) 0.8 / 334 0.76/336/4.61 0.7 / 314 0.8 / 336 0.76/336/0.09 0.75/334/6.16 1bx6:_(337) 1.0 / 334 1.01/336/4.67 1.0 / 314 1.0 / 336 1.01/336/0.43 1.01/334/6.09 1stc:E(334) 1.1 / 334 1.1/336/4.98 1.1 / 333 1.1 / 334 1.1/336/0.07 1.09/334/5.25 1cmk:E(350) 2.0 / 335 2/336/5.82 2.0 / 331 1.5 / 316 1.72/330/0.77 1.71/330/6.72 1daw:A(327) 3.1 / 267 2.77/266/10.99 2.8 / 259 2.0 / 239 1.87/250/0.53 1.92/252/5.91 1qmz:C(296) 2.5 / 259 2.07/252/6.94 2.3 / 233 1.9 / 239 1.9/251/0.67 1.96/253/5.38 1day:A(327) 2.7 / 263 2.61/262/9.96 2.9 / 262 2.0 / 239 1.96/252/0.49 1.98/253/5.99 1koa:_(447) 2.8 / 261 2.7/258/10.81 2.4 / 225 2.1 / 233 2.16/249/0.42 2.14/249/8.26 1jnk:_(346) 2.8 / 253 2.49/194/11.97 3.0 / 240 2.2 / 220 2.19/242/0.29 2.23/244/6.21 1gag:A(300) 2.8 / 265 2.87/267/7.78 2.7 / 247 2.3 / 232 2.36/251/0.67 2.46/254/5.37 1bl7:A(351) 3.5 / 254 3.14/246/8.79 3.1 / 2232.3 / 220 2.39/235/0.54 2.4/236/6.33 1cja:B(327) 4.7 / 165 4.19/165/10.9 - 2.7 / 115 2.85/143/0.483.01/149/6.36 1e7v:A(850)4.0 / 159 4.43/165/43.51 - 2.8 / 116 3/142/0.75 3.1/155/15.59 1bo1:B(318) 3.9 / 138 3.9/145/12.25 - 3.0 / 103 2.98/136/0.16 2.9/135/5.71 1b40:A(517) 3.4 / 45 5.68/83/21.4 - 2.9 / 57 3/107/0.67 3.36/105/9.44 1lar:B(533)2.6 / 345.77/123/23.11-3.0 / 663.07/88/0.793.21/86/10.1410 difficult casesIn this experiment, we experiment on a well-known data set, 10 difficult cases 25 reported by Fisher, 1996. Table 2 displays all the structure alignment results for 10 difficult cases. The EMPSC performs worse in case (1ten:_(89) vs. 3hhr:B(195))Table 2. Comparison of different structure alignment results for 10 difficult cases10 difficult cases (Fisher 1996)Dali CE V AST ProSup FLASH EMPSC Protein 1 (residues) Protein 2 (residues) rmsd/#res rmsd/#res/sec rmsd/#res rmsd/#res rmsd/#res/sec rmsd/#res/sec1bge:B(159) 2gmf:A(121) 3.3 / 94 4.02/102/2.59 2.3 / 71 2.4 / 87 -/-/-a2.56/95/0.441cew:I(108) 1mol:A(94) 2.3 / 81 2.34/81/2.07 2.0 / 71 1.9 / 76 1.92/79/0.07 2.11/81/0.49 1cid:_(177) 2rhe:_(114) 3.1 / 96 2.97/98/2.4 2.0 / 78 2.3 / 84 2.24/94/0.24 2.23/94/1.191crl:_(534) 1ede:_(310) 3.6 / 212 3.91/220/16.29 3.7 / 186 2.6 / 161 2.49/191/0.792.7/199/9.31fxi:A(96) 1ubq:_(76) 2.5 / 52 2.79/64/1.79 2.1 / 48 2.6 / 54 2.47/62/0.03 2.56/63/0.471ten:_(89) 3hhr:B(195) 1.9 / 86 1.9/87/2.14 1.5 / 76 1.7 / 85 1.73/86/0.21 2.2/76/1.01 1tie:_(166) 4fgf:_(124) 3.1 / 114 2.86/115/2.23 1.6 / 76 2.4 / 104 2.28/108/0.29 2.44/113/1.152sim:_(381) 1nsb:A(390) 3.2 / 289 2.99/276/9.24 4.2 / 299 2.6 / 248 2.61/276/7.8 2.71/282/8.962aza:A(129)1paz:_(120)3.0 / 822.9/85/1.942.1 / 702.6 / 822.34/81/0.1 2.22/82/0.883hla:B(99) 2rhe:_(114) 3.0 / 74 3.46/85/2.49 2.3 / 58 2.7 / 71 2.94/79/0.09 2.75/77/0.65a This result is available in the original FLASH paper, but we could not get any result while running the FLASH program provided by the authors.Special cases in global alignment – dissimilar protein comparisonsIn this experiment, we experiments on dissimilar but comparable proteins. If two proteins are dissimilar or quite different in same family, EMPSC can still obtain better results than ProSup and FLASH methods. There proteins are selected from the first experiment, because they belong to the same family. Table 3 displays the proteins that we selected and the results. According to these results, EMPSC performs comparablely both in number of corresponding residues and r.m.s.d.The “–” notation in FLASH column of Table 3 represents that FLASH didn’t find any statistical significant solutions. As FLASH uses the statistical method to increase its computing speed in choosing candidates, in similar cases of proteins, it can get very good solutions. But in dissimilar cases of proteins, the statistical significance evaluation will reject further processing. Generally speaking, only globally similar proteins could pass the statistical significance measurement. Even though, there is some significant local alignment information, it will not process. Comparing the global structure from local structure alignment is the advantage of EMPSC algorithm. EMPSC algorithm performs the worst in the following three cases – (1e7v:A(850) vs. 1jnk:_(346)),(1e7v:A(850) vs. 1day:A(327)) and (1bo1:B(318) vs. 1day:A(327)).Table parison of dissimilar cases of proteins with ProSup, FLASH, and EMFLASH EMPSCCE ProSuprmsd/#res/sec rmsd/#res Rmsd/#res/sec rmsd/#res/sec1cja:B(327) 1daw:A(327) 3.73/157/8.15 3.0 / 131 -/-/-a 3.02/153/5.831cja:B(327) 1qmz:C(296) 3.58/153/7.28 3.0 / 133 2.81/150/0.39 2.83/151/5.261cja:B(327) 1day:A(327) 3.79/157/8.14 3.1 / 131 -/-/- 3.04/152/5.711cja:B(327) 1koa:_(447) 4.6/173/15.81 3.0 / 130 2.92/150/0.8 3.14/160/7.91cja:B(327) 1jnk:_(346) 4.24/162/12.93 2.9 / 133 3.09/140/0.83 3.07/157/6.221e7v:A(850) 1daw:A(327) 3.59/135/28.94 3.1 / 110 3.19/147/1.93 2.99/144/15.021e7v:A(850) 1qmz:C(296) 4.34/156/45.17 2.8 / 119 2.78/139/1.5 3.15/147/13.611e7v:A(850) 1day:A(327) 4.25/148/42.39 3.0 / 110 -/-/- 3.48/97/14.921e7v:A(850) 1koa:_(447) 4.31/163/77.09 3.2 / 115 2.99/146/1.1 3.17/144/20.841e7v:A(850) 1jnk:_(346) 3.91/143/58.76 3.0 / 120 2.89/152/1.31 3.23/88/15.931bo1:B(318) 1daw:A(327) 3.75/144/9.01 3.1 / 121 3.13/129/0.39 2.86/136/5.581bo1:B(318) 1qmz:C(296) 3.6/145/8.55 2.8 / 123 2.72/133/0.28 2.71/134/4.991bo1:B(318) 1day:A(327) 4.04/151/10.05 2.9 / 123 2.85/139/0.39 3.4/100/5.521bo1:B(318) 1koa:_(447) 4.03/146/16.972.8 / 118 -/-/- 2.81/130/7.77 1bo1:B(318) 1jnk:_(346)3.96/151/16.282.8 / 114 -/-/-3.21/139/5.9 a “─” represents that FLASH doesn’t provide any solution.DiscussionEfficiency and Number of Alternative SolutionsAlthough, in previous section, we listed the execution time of every comparison In Table 1, Table 2 and Table 3, it is very hard to observe the performance relationship between CE, FLASH and EMPSC. Therefore, we added the reside numbers of the two compared proteins, and plot the relationship diagram of execution time vs. total residues, as Fig 6. In this figure, it is obviously that EMPSC is truly faster than CE, especially for large protein structure comparisons. However, EMPSC looksslower than FLASH.10203040506070809002004006008001000120014001600Total Residues of Compared Proteins E x e c u t i o nT i m e (s e c o n d s )Fig 6. The execution time of CE, FLASH and EMPSC, given different total residues of compared proteins. These trend lines for each method are polynomial regressions of order 2 which are provided by MS Excel Trendfunction.In order to know whether we can further speed up EMPSC, we did more experiments about EMPSC with different numbers of alternative solutions. We repeated the experiments in previous section with Top-3 and Top-5 alternative solutions and compared it with Top-10 results and FLASH. The detail results listed as Table 4 in appendix section, and we plot the diagram of the execution time versusdifferent number of alternative solutions, as Fig 7. In this diagram, we can found that execution time of EMPSC is perfectly proportional to the number of alternative solutions. After profiling our EMPSC program, we found EMPSC spend most execution time in alignment refining process. And we found that, if FLASH provides alternative solutions, it spends about the same execution time as EMPSC. Two data points of FLASH in Fig 7 show such case. This conclusion can be applied in any PSC algorithm that claims fast but provides only one solution (like FAST 15), except hash-based alignment refining algorithm.Obviously, according to our observation, EMPSC is a good choice for solving protein structure comparison problems. In addition, we can conclude that further enhancement of PSC algorithmsshould be focused on the alignment refining process.510152025303502004006008001000120014001600Total Residues of Compared ProteinsE x e c u t i o n T i m e (s e c o n d s )Fig 7. The execution time of FLASH and EMPSC Top-3, Top-5, Top-10 alternative solutions. These trend lines for each method are polynomial regressions of order 2 which are provided by MS Excel Trend function. Characteristic of EMPSC AlgorithmThe proposed EMPSC algorithm possesses three major features, which we believe that these features make EMPSC a good choice of PSC algorithms. First, the ellipsoidal representation can provide a good summary of 3D information for residues segment. Particularly, two ellipsoidal models can easily map to each other via transformations or rotations of coordination systems. As we said in introduction, to represent α-helix in a single vector is proper, because the α-helix is hard to bend. The representing vector of β-sheet does drop some structure information. As the β-sheet structure is usually bending or。

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托福听力tpo50 lecture1、2、3、4 原文+题目+答案+译文

托福听力tpo50 lecture1、2、3、4 原文+题目+答案+译文

托福听力tpo50 lecture1、2、3、4 原文+题目+答案+译文Lecture1 (1)原文 (1)题目 (4)答案 (6)译文 (6)Lecture2 (8)原文 (8)题目 (10)答案 (12)译文 (12)Lecture3 (14)原文 (14)题目 (16)答案 (18)译文 (18)Lecture4 (20)原文 (20)题目 (22)答案 (24)译文 (24)Lecture1原文NARRATOR: Listen to part of a lecture in an ancient history class.FEMALE PROFESSOR: OK, last time we were discussing trade and commerce during the Bronze Age … And I said a little over 3,000 years ago there was quite a lively trade among the countries along the Mediterranean Sea—people were making objects out of bronze, and they were using bronze tools to make other goods, and they developed trade networks to trade these goods with other countries around the Mediterranean … One of the things they traded was glass …And recently there was an archeological excavation in Egypt—on the Nile River, around where it enters the Mediterranean Sea—where they discovered an ancient glass factory. Robert?MALE STUDENT: I thought our textbook said that the Egyptians imported their glass from other countries.FEMALE PROFESSOR: Well, until now that's what the evidence seemed to suggest. I mean, we had some evidence that suggested that the Egyptians were making glass objects, uh, but not glass.MALE STUDENT: OK, am-am I missing something? They're making glass, but they're not making glass.FEMALE PROFESSOR: I said they were making glass objects, right? You see, it was previously thought that they weren't actually making the raw glass itself, that they were importing unfinished glass from Mesopotamia—um, which today is a region consisting of Iraq, and parts of Syria, Turkey, and Iran—and simply reworking it. Most archeologists believed that the glass factories were in Mesopotamia because that's where the oldest known glass remains come from. You see, there were two stages of glassmaking: the primary production stage, where they made disks of raw glass… Uh, an- and then there was the secondary stage, where they melted the raw glass, the glass disks, and created decorative objects or whatever.And from this new Egyptian site we've learned that the primary production stage had several steps. First, they took quartz—a colorless, transparent mineral—and crushed it. Then they took that crushed quartz and mixed it with plant ash; uh, “plant ash” is just what it soundslike—the ash that's left after you've burned plant material. They slowly heated this mixture, at a relatively low temperature, in small vessels, um containers, like jars, made out of clay. Uh, and that yielded a kind of glassy material…They took this glassy material and ground it up into a powder, and then they used metallic dye to color it… After that, they poured the colored powder out into disk-shaped molds and heated it up to very high temperatures, so that it melted. After it cooled, they'd break the molds, and inside…there were the glass disks. These disks were shipped off to other sites within Egypt and places around the Mediterranean. Then, in the secondary phase, the disks were reheated and shaped into decorative objects. Susan?FEMALE STUDENT: So what kind of objects were people making back then? FEMALE PROFESSOR: Well, the most common objects we’ve found—mostly in Egypt and Mesopotamia—uh, the most common objects were beads; one thing Egyptians were very, very good at was imitating precious stones; they created some beads that looked so much like emeralds and pearls that it was very difficult to distinguish them from the real thing. Uh, and-and also beautiful vessels, uh, with narrow necks; they were probably really valuable, so they wouldn't have been used to hold cooking oil or common food items; they were most likely used for expensive liquids like perfume. Now the glass made at this factory was mostly red; to get this red color, they used copper; in a sophisticated process. Of course, any kind of glass was very valuable, so these red bottles would only have been owned by wealthy people. In fact, because it was so difficult to make, and sort of mysterious and complicated, it was probably a product produced for the royal family, and they probably used glass to show their power. Also, beautiful, expensive objects make great gifts if you're looking to establish or strengthen political alliances…and it's quite possible that ancient Egyptians were actually exporting glass, not just making it or importing it. The trade with Mesopotamia was probably a friendly, mutual trade…because, uh, Mesopotamian glass was usually white or yellow, so Mesopotamians might have said something like, “We'll give you two white disks for two red disks.” There’s no proof ofthat, uh—at least not yet…题目1.What is the lecture mainly about?A. New information about glass production and use in ancient EgyptB. Whether Egyptians or Mesopotamians were the first to invent glassC. Differences between Egyptian glass and other kinds of glassD. Reasons why ancient Egyptians imported glass from other countries2.What is the importance of the archaeological evidence recently found in Egypt?A. It supports the theory that ancient Egyptians imported glass from Mesopotamia.B. It proves that ancient Egyptians made glass objects prior to the Bronze Age.C. It provides the first evidence that glassmaking in the Bronze Age required two different stages.D. It shows that ancient Egyptians were producing raw glass.3.The professor describes a process for making glass disks. Summarize the process by putting the steps in the correct order. [Click on a sentence. Then drag it to the space where it belongs. The last one is done for you.]A.Glass-like material is ground up and dyed blue or red.B.Powdered material is heated at very high temperatures.C.Crushed quartz and plant ash are heated at low temperatures.D.Containers are broken to remove glass disks.4.Based on the lecture, what are two kinds of glass objects that were valued in ancient Egypt and Mesopotamia? [Click on 2 answers.]A. BeadsB. Cooking utensilsC. ContainersD. Windows5.According to the professor, what are two reasons why ancient Egyptians exported glass? [Click on 2 answers.]A. To build relationships with foreign leadersB. To hold cooking oil that was sold in other countriesC. To get bronze tools from other countriesD. To acquire colors of glass not made in Egypt6.Why does the professor say this:Robert: Ok. Am……Am I missing something? They are making glass but they are not making glass?Professor: I said they were making glass objects, right?A. To emphasize that glass objects were only made in ancient EgyptB. To find out what the student does not understandC. To indicate that there was no contradiction in her previous statementD. To correct what she said in her previous statement答案A D CABD AC AD C译文旁白：请听一个古代历史课上的讲座片段。

characterization method

characterization method

Characterization MethodIntroductionCharacterization method refers to the process of determining and analyzing the properties, qualities, and features of a particular object or system. It plays a crucial role in various fields, including materials science, chemistry, biology, and engineering. By employing appropriate characterization techniques, researchers can gain valuable insights into the structure, composition, behavior, and performance of different materials and systems. This article aims to provide an overview of different characterization methods commonly used inscientific research.Types of Characterization MethodsCharacterization methods can be broadly categorized into two types: physical characterization methods and chemical characterization methods.1. Physical Characterization MethodsPhysical characterization methods focus on examining the physical properties and behavior of materials. Some commonly used physical characterization techniques include:•Microscopy: Microscopy techniques such as optical microscopy, electron microscopy (scanning electron microscopy (SEM) andtransmission electron microscopy (TEM)), atomic force microscopy(AFM), and scanning probe microscopy (SPM) enable researchers tovisualize and study the microstructure of materials at differentscales.•Spectroscopy: Spectroscopic techniques involve the interaction of materials with electromagnetic radiation to analyze theircomposition and structure. Examples include infrared spectroscopy (IR), Raman spectroscopy, ultraviolet-visible spectroscopy (UV-Vis), X-ray photoelectron spectroscopy (XPS), nuclear magneticresonance spectroscopy (NMR), and mass spectrometry.•X-ray Diffraction: X-ray diffraction (XRD) is a powerful technique for determining the crystal structure of materials byanalyzing the diffraction patterns produced when X-rays interactwith a sample. It provides information about crystal phaseidentification, lattice parameters, crystallite size, textureanalysis, etc.•Thermal Analysis: Thermal analysis techniques such asdifferential scanning calorimetry (DSC) and thermogravimetricanalysis (TGA) measure changes in material properties as afunction of temperature. These methods help in understanding phase transitions, thermal stability, decomposition behavior, and otherthermal properties of materials.•Mechanical Testing: Mechanical testing methods, including tensile testing, compression testing, hardness testing, and impact testing, evaluate the mechanical properties of materials such as strength,elasticity, hardness, toughness, and fatigue resistance.2. Chemical Characterization MethodsChemical characterization methods focus on identifying and analyzing the chemical composition of materials. Some commonly used chemical characterization techniques include:•Chromatography: Chromatographic techniques such as gaschromatography (GC), liquid chromatography (LC), and high-performance liquid chromatography (HPLC) separate and analyze thecomponents of a mixture based on their differential affinities fora stationary phase and a mobile phase.•Mass Spectrometry: Mass spectrometry (MS) is a technique that ionizes molecules and measures their mass-to-charge ratio. Itprovides information about the molecular weight, structure,composition, and fragmentation pattern of compounds.•Nuclear Magnetic Resonance: Nuclear magnetic resonance (NMR) spectroscopy detects the magnetic properties of atomic nuclei todetermine molecular structures. It is widely used for structuralelucidation and identification of organic compounds.•Electrochemical Analysis: Electrochemical techniques such as cyclic voltammetry (CV), electrochemical impedance spectroscopy(EIS), and potentiostatic/galvanostatic measurements investigatethe electrochemical behavior and properties of materials invarious environments.Application AreasCharacterization methods find applications in numerous scientific disciplines and industries. Some notable application areas include:1.Materials Science: Characterization methods play a vital role instudying the structure-property relationships of materials. Theyare used to analyze material composition, crystal structuredetermination, defect analysis, surface morphology examination,and mechanical property evaluation.2.Chemistry: Chemical characterization techniques are extensivelyemployed in organic and inorganic chemistry for compoundidentification, purity analysis, reaction monitoring, andstructure elucidation.3.Biology: Characterization methods help biologists understand thestructure and function of biological molecules such as proteins,nucleic acids, and carbohydrates. Techniques like X-raycrystallography, NMR spectroscopy, and mass spectrometry aid instudying biomolecular structures and interactions.4.Pharmaceuticals: Characterization methods are crucial in drugdevelopment and quality control processes. They ensure the purity, stability, and efficacy of pharmaceutical compounds by analyzingtheir chemical composition, crystallinity, dissolution behavior,and formulation characteristics.5.Environmental Science: Characterization methods are used toanalyze pollutants in air, water, soil samples to assessenvironmental quality. Techniques like gas chromatography-massspectrometry (GC-MS), atomic absorption spectroscopy (AAS), and X-ray fluorescence (XRF) help identify and quantify contaminants.ConclusionCharacterization methods provide valuable insights into the properties, composition, behavior, and performance of materials and systems across various scientific disciplines. By employing a combination of physical and chemical characterization techniques, researchers can gain a comprehensive understanding of their samples. These insights drive advancements in fields such as materials science, chemistry, biology, pharmaceuticals, and environmental science. Continuous development ofnew characterization methods further enhances our ability to explore the microscopic world and unravel the mysteries of nature.Note: The content provided above is a general overview of characterization methods. It is not an exhaustive list or detailed explanation of each technique.。

生物学英语复试题及答案

生物学英语复试题及答案

生物学英语复试题及答案一、选择题（每题2分，共20分）1. Which of the following is not a characteristic of living organisms?A. ReproductionB. GrowthC. Response to stimuliD. Inanimate objects2. What is the basic unit of life?A. CellB. OrganC. TissueD. Organ system3. What is the process by which organisms convert sunlight into chemical energy?A. RespirationB. PhotosynthesisC. FermentationD. Cellular respiration4. Which of the following is not a type of symbiotic relationship?A. MutualismB. CommensalismC. ParasitismD. Competition5. What is the term for the study of the structure of organisms?A. AnatomyB. PhysiologyC. EcologyD. Taxonomy6. What is the primary function of the mitochondria in a cell?A. Protein synthesisB. DNA replicationC. Energy productionD. Cell wall synthesis7. What is the term for the process by which new speciesevolve from existing ones?A. AdaptationB. Natural selectionC. SpeciationD. Genetic drift8. Which of the following is not a type of genetic mutation?A. DeletionB. InsertionC. DuplicationD. Mitosis9. What is the term for the study of the diversity of life on Earth?A. BiodiversityB. BiotechnologyC. BioinformaticsD. Biogeography10. What is the process by which organisms obtain nutrients from their environment?A. IngestionB. AssimilationC. DigestionD. Absorption二、填空题（每题2分，共20分）1. The scientific method involves making observations, forming hypotheses, conducting experiments, and then drawing ________.2. The genetic material in cells is composed of molecules called ________.3. The process by which organisms produce offspring that are similar to themselves is known as ________.4. In an ecosystem, the transfer of energy from one trophic level to the next is called ________.5. The study of the interactions between organisms and their environment is known as ________.6. The smallest unit of matter that retains the properties of an element is called a(n) ________.7. The process by which plants absorb water and nutrients from the soil is called ________.8. The study of the classification of organisms based ontheir evolutionary relationships is known as ________.9. The process by which organisms break down complex organic molecules into simpler ones is called ________.10. The study of the nervous system, including the brain,spinal cord, and nerves, is known as ________.三、简答题（每题10分，共40分）1. Explain the difference between asexual and sexual reproduction.2. Describe the role of DNA in the inheritance of traits.3. Discuss the importance of biodiversity for ecosystems.4. Explain how natural selection contributes to the evolution of species.四、论述题（20分）Discuss the impact of genetic engineering on modern agriculture and its potential ethical concerns.答案：一、选择题1. D2. A3. B4. D5. A6. C7. C8. D9. A10. A二、填空题1. Conclusions2. DNA3. Reproduction4. Energy flow5. Ecology6. Atom7. Absorption8. Phylogenetics9. Catabolism10. Neuroscience三、简答题1. Asexual reproduction involves a single organism producing offspring that are genetically identical to itself, while sexual reproduction involves the combination of genetic material from two organisms to produce offspring with a mix of traits from both parents.2. DNA carries the genetic information that determines the traits of an organism. It is passed down from parents to offspring during reproduction, allowing for the inheritance of specific characteristics.3. Biodiversity is crucial for ecosystems as it contributes to their stability, resilience, and ability to adapt to changes. It also supports a wide range of ecosystem services that are essential for human well-being.4. Natural selection is the process by which organisms that are better adapted to their environment are more likely to survive and reproduce, passing on their advantageous traits to their offspring. Over time, this leads to the evolution of species as traits that enhance survival and reproduction become more common.四、论述题Genetic engineering has revolutionized modern agriculture byenabling the development of crops with desirable traits such as disease resistance, pest resistance, and higher yields. However, it also raises ethical concerns regarding the potential impact on biodiversity, the possibility of creating "superweeds" or "superbugs," and the long-term health effects of genetically modified organisms on humans and the environment. It。

Euler’sMethod

Euler’sMethod

Euler’s MethodEuler’s method is a numerical method for solving initial value problems.Euler’s method is based on the insight that some differential equations(which are the ones we can solve using Euler’s method)provide us with the slope of the function(at all points),while an initial value provides us with a point on the ing this information we can approximate the function with a tangent line at the point given by the initial value.As we have seen,the tangent line is only a good approximation over a small interval.Thus,after moving a small interval,we will want to construct a new tangent line.What happens when we move to the new point?Although we know the value of the deriva-tive of the function at the new point,we do not know the exact value of the function.Thus, the best we can do is construct an approximate tangent line,using the actual slope of the function,and an approximation to the value of the function at point of tangency;the approx-imate value of the function is given by the previous tangent line approximation.Continuing in this fashion,we construct a piecewise-linear approximation to the solution of the differ-ential equation.We can also think of our approximation as a discrete function,which is defined for these approximated points.To make this discrete function a continuous function, we interpolate linearly between each pair of these points(draw a line between them). Before we begin we should emphasize the limitations of Euler’s method are practical,not theoretical.In theory,we should be able to make the distance traveled along each tangent line as short as we like,making the approximation to the actual solution of the initial value problem as accurate as we like.The only reason we cannot do this in practice is because there is a limit to the number of computations we can perform.From a human perspective, it becomes unreasonable to look at more than10-100intervals.A computer might do well with billions of intervals.Even still,we would need to look in a limit as the length of the intervals approaches0,which means the number of intervals approaches∞.Suppose we wish to solve the initial value problemdm=f(t)dtwith initial condition m(t0)=m0.Our goal is to construct a functionˆm(t)which approxi-mates the actual solution.First we setˆm(t0)=m(t0)=m0.We will defineˆm(t)on intervals of∆t,so it will next be defined atˆm(t0+∆t),thenˆm(t0+2∆t),and so on.If we haveˆm(t) wefindˆm(t+∆t)by moving along the line with slope m (t)(the slope of the line tangent to m(t))passing through the point(t,ˆm(t))(which is an approximation of the point the tangent line really intersects).Doing so wefind thatˆm(t+∆t)=ˆm(t)+m (t)∆twhere m (t)is given by the differential equation in the initial value problem.This is a discrete function which can be made continuous through linear interpolation.This continous function is defined on intervals,and for any point in the interval[t0,t0+∆t]thefunction is given byˆm(t0+t)=ˆm(t0)+m (t0)twhere t∈[0,∆t](this restriction upon t ensures that t0+t∈[t0,t0+∆t]).This is exactly the equation of the approximate tangent line to the function m(t)at the point t0(called the approximate tangent line because we useˆm(t0)rather than m(t0),which is an unknown,for our base point).By substituting∆t into this equation we can verifyˆm(t0+∆t)=ˆm(t0)+m (t0)∆twhich is consistent with our discretely definedˆm(t).Although this is an iterative process, we can move directly to the solution after multiple intervals quite easily.For instance,ˆm(t0+4∆t)=ˆm(t0)+m (t0)∆t+m (t0+∆t)∆t+m (t0+2∆t)∆t+m (t0+3∆t)∆t where we simply move along each of the approximate tangent lines all in one step.It is often easiest to solve problems of this type using a chart.We will outline this process in the following examples.Example1Apply Euler’s method to the differential equationdV=2tdtwithin initial condition V(0)=2.Approximate the value of V(1)using∆t=0.25. Solution We begin by settingˆV(0)=2.Next we construct the charttˆV(t)V (t)=2tˆV(t+∆t)=ˆV(t)+V (t)∆t022·0=02+0·0.25=20.2522·0.25=0.52+0.5·0.25=2.1250.52.1252·0.5=12.125+1·0.25=2.3750.752.3752·0.75=1.52.375+1.5·0.25=2.7512.75So wefind that V(1)≈2.75.It turns out that V(t)=t2+2is the analytical solution to this differential equation,and V(1)=3,so Euler’s method provides a somewhat reasonable approximation,which could be greatly improved upon by decreasing the size of∆t.Example2Apply Euler’s method to the differential equationdP=e−tdtwith the initial condition P(0)=0.Approximate the value of P(2).Solution We begin by settingˆP(0)=0.We will use the time step∆t=0.5.Next we construct the charttˆP(t)P (t)=e−tˆP(t+∆t)=ˆP(t)+P (t)∆t00e−0=10+1·0.5=0.50.50.5e−0.5≈0.606530.5+0.60653·0.5=0.8032610.80326e−1≈0.367880.80326+0.36788·0.5=0.987211.50.9872e−1.5≈0.223130.9872+.22313·0.5=1.0987721.09877The analytical solution to this problem is P(t)=1−e−t,so wefind the exact value P(2)=0.86466,which is somewhat lower than our approximation.By shrinking the size of the interval∆t,we could calculate a more accurate estimate.In thefirst case above,we know the analytical solution is t2+2,which is concave up.The fact that the function is concave up means that the value for the derivative at the beginning of an interval is smaller than the value for the derivative at the end of the interval.The result is that Euler’s method continually underestimates the actual value of the result.In the second case,our function1−e−t is concave down,so the value of the derivative is decreas-ing,and the derivative at the beginning of an interval is always larger than at the end.For this reason,our Euler’s method approximation continues to overestimate the actual function.One might think to remedy this situation by using the slope at the end of an interval,rather than at the beginning.Let’s examine thefirst initial value problem,and make approxima-tions using the slope at the end of each interval(note that when we do this our method has less and less resemblence to tangent line approximations).Example3Apply Euler’s method(using the slope at the right end points)to the differential equationdV=2tdtwithin initial condition V(0)=2.Approximate the value of V(1)using∆t=0.25. Solution We begin by settingˆV(0)=2.Next we construct the charttˆV(t)V (t+∆t)=2tˆV(t+∆t)=ˆV(t)+V (t+∆t)∆t022·0.25=0.52+0.5·0.25=2.1250.2522·0.5=12.125+1·0.25=2.3750.52.1252·0.75=1.52.375+1.5·0.25=2.750.752.3752·1=22.75+2·0.25=3.2513.25So wefind that V(1)≈3.25.As above,V(t)=t2+2is the analytical solution to this differential equation,and V(1)=3,so in this situation our estimate is high.If we were to continue with smaller and smaller∆t intervals,we could get a better approx-imation from above and below.In truth,however,there are much better-suited algorithms to numerically solving differential equations.Most of these methods use the same basic idea as Euler’s method,but alter the process in order to make the method more accurate for thenumber of intervals used.For instance,the Runge-Kutta method uses weighted averages of the slope of the function at multiple points in a given interval(for instance,we may take a weighted average of the slope at the beginning,end,and midpoint of an interval,in order to estimate the next value of our approximate solution).We will not go into the details of actually solving differential equations using the Runge-Kutta method,but suffice ourselves to say that the method is conceptually similar to Euler’s method.The above examples may seem somewhat artificial,in that we already knew an analytic solution to the differential equation(which we canfind using the indefinite integral).The truth of matter is,however,that many differential equations do not have a closed-form, analytic solution(which means there is no combination of elementary functions that when differentiated satisfy the differential equation).In some situations,we can get around this by looking at infinite series.However,if we can’t use some special trick,then the best we can do is look for a numerical solution.This is when the true utility of numerical methods shines.One example of such a function is1√2πe−x22which corresponds to a Gaussian,or normal distribution(such functions are ubiquitous in statistics,because they have a number of nice properties:the mean,median,and mode are the same,there is symmetry around the mean,and inflection points occurs one standard deviation away from the mean).This function can be integrated by using infinite series,but we are a long way away from being able to use any such technique.Instead,we can use Euler’s method to atleast approximate a solution.Example4Apply Euler’s method(using the slope at the right end points)to the differential equationd f dt =1√2πe−t22within initial condition f(0)=0.5.Approximate the value of f(1)using∆t=0.25. Solution We begin by settingˆf(0)=0.5.We will use the time step∆t=0.25.Next we construct the charttˆf(t)f (t)=1√2πe−t22ˆf(t+∆t)=ˆf(t)+f (t)∆t00.50.3990.5+0.399·0.25=0.599750.250.599750.38660.59975+0.3866·0.25=0.69640.50.69640.3520.6964+0.352·0.25=0.78440.750.78440.3011370.7844+0.301137·0.25=0.85968410.859684Right now we can’t verify this result,but with a little knowledge of statistics,we know that we should get a result of about0.84.In this situation,Euler’s method provides us with a very accurate estimate.We can also used the idea of Euler’s method when we do not have a differential equation, but simply data.Suppose we have measured the velocity of an object at various times,and want to estimate the position.We know position is given by the differential equationdx=v(t)dtso we can perform Euler’s method,but rather than evaluating v(t)at the points of interest, we would just look up the value in a table.Example5Suppose you have the following data for the velocity of an object.Find the distance it travels after60seconds.Time(seconds)Velocity(m/s)001012201830154012508600Solution Here we can construct our table as normal,simply using the values we already know for velocity.Since we are looking at distance traveled,we will let x(0)ˆx(0)=0.Our data comes in increments of10,so we will let∆t=10.tˆx(t)v(t)ˆx(t+∆t)=ˆx(t)+v(t)∆t0000+0·10=0100120+12·10=1202012018120+18·10=3003030015300+15·10=4504045012450+12·10=570505708570+8·10=660606600So wefind that the object has traveled a total of660meters in1minute.Note that if we are only interested in the object’s position at t=60,we can simply use the shortcutˆx(60)=ˆx(0)+10(v(10)+v(20)+v(30)+v(40)+v(50))Finally,if we had uneven intervals of time,we could use the formulaˆx(t+∆t1+∆t2)=ˆx(t)+v(t)·∆t1+v(t+∆t1)·∆t2。