MAPLE

 

Introduction to MAPLE
Maple, first released in 1981 byWaterloo Maple,Inc., http://www.maplesoft.com/

is a system for doing mathematics on a computer. Maple combines symbolic

manipulation, numerical mathematics, outstanding graphics, and a sophisticated programming language. Because of its versatility, Maple has established itself as the computer algebra system of choice for many computer users including commercial and government scientists and engineers  mathematics, science, and engineering teachers and researchers, and students enrolled in mathematics, science, and     
engineering courses. However, due to its special nature and sophistication, beginning users need to be aware of the special syntax required to make Maple perform in the way intended. You will find that calculations and sequences of calculations most frequently used by beginning users are discussed in detail along with many typical examples. In addition, the comprehensive index not only lists a variety of topics but also cross-references commands with frequently used options. Maple by Example serves as a valuable tool and reference to the beginning user of Maple as well as to the more sophisticated user, with specialized needs.

Getting Started with Maple

We begin by introducing the essentials of Maple. The examples presented are

taken from algebra, trigonometry, and calculus topics that you are familiar with to

assist you in becoming acquainted with the Maple computer algebra system.

Five Basic Rules of Maple Syntax

1. The arguments of all functions (both built-in ones and ones that you define) are given in parentheses (...). Brackets [...] are used for grouping operations:

vectors, matrices, and lists are given in brackets.

2. A semicolon (;) or colon (:) must be included at the end of each command.

Maple does not display the result when a colon is included at the end of a

command. Never name a user-defined object with the same name as that of a

built-in Maple object.

3. Multiplication is represented by an asterisk, *. Enter 2*x*y to evaluate 2xy

not 2xy.

4. Powers are denoted by a ˆ. Enter (8*xˆ3)ˆ(1/3) to evaluate (8x3)1/3 =

81/3(x3)1/3 = 2x instead of 8*xˆ1/3, which returns 8x/3.

5. Maple follows the order of operations exactly. Thus, entering (1+x)ˆ1/x

returns (1+x)1

x while (1+x)ˆ(1/x) returns (1+x)1/x. Similarly, entering xˆ3*x

returns x3 · x = x4 while entering xˆ(3*x) returns x3x.

Remark: If you get no response or an incorrect response, you may have entered or executed the command incorrectly. In some cases, the amount of memory allocated to Maple can cause a crash. Like people, Maple is not perfect and errors can occur.

SCATTER PLOT IN EXCEL

Beer's Law 

Beer's Law states that there is a linear relationship between concentration of a colored compound in solution and the light absorption of the solution. This fact can be used to calculate the concentration of unknown solutions, given their absorption readings. First, a series of solutions of known concentration are tested for their absorption level. Next, a scatter plot is made of this empirical data and a linear regression line is fitted to the data. This regression line can be expressed as a formula and used to calculate the concentration of unknown solutions.

Titration Data Plotting

In this next part of the tutorial, we will work with another set of data. In this case, it is of a strong acid-strong base titration. With this titration, a strong base (NaOH) of known concentration is added to a strong acid (also of known concentration, in this case). As the strong base is added to solution, its OH- ions bind with the free H+ions of the acid. An equivalence point is reached when there are no free OH- nor H+ ions in the solution. This equivalence point can be found with a color indicator in the solution or through a pH titration curve. This part of the tutorial will show you how to do the latter.

The linear equation shown on the chart represents the relationship between Concentration (x) and Absorbance (y) for the compound in solution. The regression line can be considered an acceptable estimation of the true relationship between concentration and absorbance. We have been given the absorbance readings for two solutions of unknown concentrationUsing the linear equation, a spreadsheet cell can have an equation associated with it to do the calculation for us. We have a value for y (Absorbance) and need to solve for x (Concentration).

• Enter and format data in an Excel spreadsheet in a form appropriate for graphing
• Create a scatter plot from spreadsheet data
• Insert a linear regression line (trendline) into the scatter plot
• Use the slope/intercept formula for the regression line to calculate a x value for a known y value
• Explore curve fitting to scatter plot data
• Create a connected point (line) graph
• Place a reference line in a graph 

Below are some of examples:

SMILES

Simplified molecular input line entry specification 

The simplified molecular input line entry specification or SMILES is a specification for unambiguously describing the structure of chemical molecules using short ASCII strings SMILES strings can be imported by most molecule editor for conversion back into two dimensional drawings or three dimensional models of the molecules.The term Canonical SMILES refers to the version of the SMILES specification that includes rules for ensuring that each distinct chemical molecule has a single unique SMILES representation.A common application of Canonical SMILES is for indexing and ensuring uniqueness of molecules in a database. The term Isomeric SMILES refers to the version of the SMILES specification that includes extensions to support the specification of isotopes, chirality, and configuration about double bonds. A notable feature of these rules is that they allow rigorous partialspecification of chirality. In terms of a graph-based computational procedure, SMILES is a string obtained by printing the symbol nodes encountered in a depth-first tree traversal of a chemical graph.The chemical graph is first trimmed to removehydrogen atoms and cycles are broken to turn it into a spanning tree. Where cycles have been broken, numeric suffix labels are included to indicate the connected nodes.Parentheses are used to indicate points of branching onthe tree.

-Simplified Molecular Input Line Entry System (SMILES)

-Widely used AND computationally efficient

-Uses atomic symbols and a set of intuitive rules

-Uses hydrogen-suppressed molecular graphs (HSMG)

below are some of the examples:

PDB

PROTEIN DATA BANK

The Protein Data Bank (PDB) is a repository for the 3-D structural data of large biological molecules, such as proteins and nucleic acids. (See also crystallographic database). The data, typically obtained by X-ray crystallography or NMR spectroscopy and submitted by biologists and biochemists from around the world, are freely accessible on the Internet via the websites of its member organisations (PDBe, PDBj, and RCSB). The PDB is overseen by an organization called the Worldwide Protein Data Bank, wwPDB.

The PDB is a key resource in areas of structural biology, such as structural genomics. Most major scientific journals, and some funding agencies, such as the NIH in the USA, now require scientists to submit their structure data to the PDB. If the contents of the PDB are thought of as primary data, then there are hundreds of derived (i.e., secondary) databases that categorize the data differently. For example, both SCOP and CATH categorize structures according to type of structure and assumed evolutionary relation, GO categorize structures based on genes.

The PDB originated as a grassroots effort. In 1971, Walter Hamilton of the Brookhaven National Laboratory agreed to set up the data bank at Brookhaven. Upon Hamilton's death in 1973, Tom Koeztle took over direction of the PDB. In January 1994, Joel Sussman of Israel's Weizmann Institute of Science was appointed head of the PDB. In October 1998, the PDB was transferred to the Research Collaboratory for Structural Bioinformatics (RCSB); the transfer was completed in June 1999. The new director was Helen M. Berman of Rutgers University (one of the member institutions of the RCSB). In 2003, with the formation of the wwPDB, the PDB became an international organization. The founding members are PDBe (Europe), RCSB(USA), and PDBj (Japan). The BMRB joined in 2006. Each of the four members of wwPDB can act as deposition, data processing and distribution centers for PDB data. The data processing refers to the fact that wwPDB staff review and annotates each submitted entry. The data are then automatically checked for plausibility. (The source code for this validation software has been made available to the public at no charge.

Clp P

Crystal Structure of ATP-dependent Clp protease subunit P from Francisella tularensis

PubMed Abstract: 

In ClpXP and ClpAP complexes, ClpA and ClpX use the energy of ATP hydrolysis to unfold proteins and translocate them into the self-compartmentalized ClpP protease. ClpP requires the ATPases to degrade folded or unfolded substrates, but binding of acyldepsipeptide antibiotics (ADEPs) to ClpP bypasses this requirement with unfolded proteins. We present the crystal structure of Escherichia coli ClpP bound to ADEP1 and report the structural changes underlying ClpP activation. ADEP1 binds in the hydrophobic groove that serves as the primary docking site for ClpP ATPases. Binding of ADEP1 locks the N-terminal loops of ClpP in a ?-hairpin conformation, generating a stable pore through which extended polypeptides can be threaded. This structure serves as a model for ClpP in the holoenzyme ClpAP and ClpXP complexes and provides critical information to further develop this class of antibiotics. 


Lon A

Crystal Structure  of Lon N-terminal domain

PubMed Abstract: 

Lon ATP-dependent proteases are key components of the protein quality control systems of bacterial cells and eukaryotic organelles. Eubacterial Lon proteases contain an N-terminal domain, an ATPase domain, and a protease domain, all in one polypeptide chain. The N-terminal domain is thought to be involved in substrate recognition, the ATPase domain in substrate unfolding and translocation into the protease chamber, and the protease domain in the hydrolysis of polypeptides into small peptide fragments. Like other AAA+ ATPases and self-compartmentalising proteases, Lon functions as an oligomeric complex, although the subunit stoichiometry is currently unclear. Here, we present crystal structures of truncated versions of Lon protease from Bacillus subtilis (BsLon), which reveal previously unknown architectural features of Lon complexes. Our analytical ultracentrifugation and electron microscopy show different oligomerisation of Lon proteases from two different bacterial species, Aquifex aeolicus and B. subtilis. The structure of BsLon-AP shows a hexameric complex consisting of a small part of the N-terminal domain, the ATPase, and protease domains. The structure shows the approximate arrangement of the three functional domains of Lon. It also reveals a resemblance between the architecture of Lon proteases and the bacterial proteasome-like protease HslUV. Our second structure, BsLon-N, represents the first 209 amino acids of the N-terminal domain of BsLon and consists of a globular domain, similar in structure to the E. coli Lon N-terminal domain, and an additional four-helix bundle, which is part of a predicted coiled-coil region. An unexpected dimeric interaction between BsLon-N monomers reveals the possibility that Lon complexes may be stabilised by coiled-coil interactions between neighbouring N-terminal domains. Together, BsLon-N and BsLon-AP are 36 amino acids short of offering a complete picture of a full-length Lon protease.


H tr A

Dengue Methyl Transferase bound to a SAM-


based Inhibitor

PubMed Abstract: 

Internal ribosome entry site (IRES) RNAs are elements of viral or cellular mRNAs that bypass steps of canonical eukaryotic cap-dependent translation initiation. Understanding of the structural basis of IRES mechanisms is limited, partially due to a lack of high-resolution structures of IRES RNAs bound to their cellular targets. Prompted by the universal phylogenetic conservation of the ribosomal P site, we solved the crystal structures of proposed P site binding domains from two intergenic region IRES RNAs bound to bacterial 70S ribosomes. The structures show that these IRES domains nearly perfectly mimic a tRNA mRNA interaction. However, there are clear differences in the global shape and position of this IRES domain in the intersubunit space compared to those of tRNA, supporting a mechanism for IRES action that invokes hybrid state mimicry to drive a noncanonical mode of translocation. These structures suggest how relatively small structured RNAs can manipulate complex biological machines.