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Fundamentals of Molecular Structural Biology
Fundamentals of Molecular Structural Biology
Fundamentals of Molecular Structural Biology
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Fundamentals of Molecular Structural Biology

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Fundamentals of Molecular Structural Biology reviews the mathematical and physical foundations of molecular structural biology. Based on these fundamental concepts, it then describes molecular structure and explains basic genetic mechanisms. Given the increasingly interdisciplinary nature of research, early career researchers and those shifting into an adjacent field often require a "fundamentals" book to get them up-to-speed on the foundations of a particular field. This book fills that niche.

  • Provides a current and easily digestible resource on molecular structural biology, discussing both foundations and the latest advances
  • Addresses critical issues surrounding macromolecular structures, such as structure-based drug discovery, single-particle analysis, computational molecular biology/molecular dynamic simulation, cell signaling and immune response, macromolecular assemblies, and systems biology
  • Presents discussions that ultimately lead the reader toward a more detailed understanding of the basis and origin of disease
LanguageEnglish
Release dateAug 13, 2019
ISBN9780128148563
Fundamentals of Molecular Structural Biology
Author

Subrata Pal

Subrata Pal obtained his bachelor’s and master’s degrees, both in physics, from Calcutta University. Subsequently, he pursued his predoctoral research in molecular biology and received a PhD degree from the same university in 1982. He carried out his postdoctoral research in DNA replication and gene expression at two of the premiere institutions in the USA: the National Institutes of Health, Bethesda, Maryland and Harvard Medical School, Boston Massachusetts. At Harvard, he received a Claudia Adams Barr special investigator award for basic contribution to cancer research. Professor Pal has a long teaching experience at the undergraduate and graduate levels – the areas of his teaching include physics, molecular biology and genomics.

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    Chapter 1

    Introduction—A historical perspective

    Abstract

    Biology, the study of living organisms and their vital processes, began during the Greek civilization with the classification of animals and plants. Hundreds of years later, the optical microscope revealed the internal structure of living organisms—that they were composed of cells. Subsequently, the living cell was found to be made up of atoms and molecules, the protein and nucleic acid being the two most important macromolecules. Development of three physical techniques in the 20th century—X-ray crystallography, nuclear magnetic resonance (NMR) spectroscopy, and electron crystallography—has facilitated the determination of the structures of these macromolecules at Ångstörm-level resolution. Concomitantly, classical biology evolved to molecular biology and then to molecular structural biology. Moving ahead, molecular structural biology is no longer restricted to the study of single molecules, but rapidly gaining competence to understand and predict the behavior of biological systems based on a set of molecules involved and the relationships between them.

    Keywords

    Cell; Nucleic acid; Protein; X-ray crystallography; NMR spectroscopy; Electron crystallography; Macromolecular structure; Molecular structural biology

    Man began to take interest in nature, which includes plants and animals, not in any aesthetic or philosophic sense, but only in relation to his immediate need primarily for food. The observational knowledge of nature, particularly the habits of animals and properties of plants, formed the basis of present day biological sciences.

    1.1 Biology begins as natural history

    Biology, as we all know, is the study of living organisms and their vital processes. It began as natural history which aimed to understand the whole organism in context. Natural history, which is essentially the study of plants and animals, was based on observational methods. It can be found from the earliest recorded history of biology that the Babylonians, Egyptians, Chinese, and Indians made innumerable observations in the course of their agricultural and medical practices. Yet, the accumulated biological information did not immediately and automatically lead to any rational concept or hypothesis. Rapid progress started to be made with the advent of Greek civilization.

    Greek civilization produced legendary personalities who, by virtue of their astounding insight, examined the phenomena of the natural world and made seminal contributions to natural philosophy. The most distinguished of them was Aristotle (384–322 BC) whose interests spanned all branches of knowledge including biology. He is recognized as the originator of the scientific study of life.

    Aristotle was the first to undertake a systematic classification of animals based on specific principles, some of which remain valid even today. However, he made little contribution to the study of plants. This was left to his student, Theophrastus, who is said to have done for botany what Aristotle did for zoology.

    1.2 Nature of matter

    The ancient Greek philosophers did not keep themselves confined to classification and categorization. They speculated about the nature of matter and formulated hypotheses. Democritus (470–380 BC) proposed that all matter consisted of tiny particles which were further indivisible. These particles were called atamos (Greek meaning: indivisible). The early atomists thought that the infinite universe consisted of atamos (or atoms) and void space.

    One of the earliest Greek philosophers, Thales of Miletus, held the view that the universe contained a creative force which he called physis (precursor of the term physics). Thales thought that the basic element of matter was water; on the other hand, Anaximenes believed that it was air, while Heraclitus maintained that it was fire. Empedocles combined the ideas of Thales, Anaximenes, and Heraclitus and added one of his own—earth. Accordingly, the notion emerged that all matter was made of differing amounts of fire, air, water, and earth held together by forces of attraction and repulsion.

    Physis as the creative force was accepted also by Hippocrates, the eminent Greek physician. However, members of the Hippocratic school gave little importance to the roles thought to be played by fire, air, water, and earth. Instead, they believed that all living bodies were made up of four humors (Latin meaning: liquid or fluid)—blood, phelgm, choler (yellow bile), and melancholy (black bile).

    Needless to say, for the ancient Greeks, it was not possible to verify all such speculations and hypotheses through experiments. The civilization had no dearth of creative minds, but lacked proper tools to conduct scientific investigations. The world had to wait hundreds of years before the microscope revealed the basic structure of both plants and animals.

    1.3 Microscope reveals internal structure of living organisms

    The magnifying power of segments of a glass sphere was known for nearly 2000 years. During the first century AD, glass was invented and the Romans were testing different shapes of clear glass to see if they could magnify an object. Sometime towards the end of 16th century, two Dutch spectacle-makers, Hans Jensen and his son Zacharias, invented the compound microscope by putting several lenses in a tube. However, their instruments were of little practical utility since the magnification was only around 9 × and the images were blurred.

    In the late 17th century, another Dutchman, Antonie van Leeuwenhoek, became the first man to make and use a real microscope. Using single lenses instead of their combinations, Leeuwenhoek observed a number of biological specimens including protozoans (which he called animalcules) and bacteria. His microscopes achieved remarkable magnifications up to 270 ×.

    1.4 Cell theory

    Improvements in microscopy facilitated the introduction of a new concept in the internal structure of living organisms—the cell. The credit for the first description of the cell goes to Robert Hooke, an English physicist and microscopist. Examining a slice of cork under the microscope, Hooke found air-filled compartments and introduced the term cells (cell in Latin means a small room) or pores to refer to these units. Hooke’s observations were published in 1665 in Micrographia. Nevertheless, the discovery had to wait nearly 200 years for its significance to be appreciated.

    In the meantime, microcopy continued to undergo technical improvement. One problem with the earlier microscopes was that of chromatic aberration, as a result of which the resolving power of the instruments was compromised. The problem was satisfactorily addressed by the introduction of achromatic microscopes in the 1830s.

    In 1838, German botanist Mathias Jacob Schleiden postulated that every structural element of plants is composed of cells or their products. Subsequently, in the following year, German zoologist Theodor Schwann extended the proposition to include animals. Biological science saw a rapprochement between botany and zoology. The conclusions of Schleiden and Schwann together formed the cell theory—a gigantic advance in the study of living organisms. Added to this in the 1850s was Rudolf Virchow’s aphorism omnis cellula a cellula (every cell from a preexisting cell). Now, the attention of the scientific world shifted to the living processes inside the cell.

    Together with the cell theory, two other landmark developments during the second half of the 19th century made the study of intracellular components all-the-more compelling. These developments were associated with two legendary individuals—Charles Darwin and Gregor Mendel.

    1.5 Theory of natural selection and laws of heredity

    In 1858, Charles Darwin published his theory of natural selection in On the Origin of Species by Means of Natural Selection. The crux of the theory is as follows: In the randomly varying nature, some variations are more advantageous than others. There is always a struggle for existence and those organisms which are better adjusted to their environment, even slightly, will most likely survive and transmit their advantageous traits to the next generation.

    On the other hand, Gregor Mendel, universally acclaimed as the father of genetics, carried out extensive fertilization experiments with garden peas and formulated a set of laws, known as the laws of heredity, for the transmission of trait units (later came to be known as genes) from one generation to another through reproductive mechanism. The first principle, the law of segregation, states that the hereditary units are paired in the parent and segregate during the formation of gametes. Secondly, the law of independent assortment states that each pair of units is inherited independently of all other pairs. The third tenet, the law of dominance, maintains that the trait units act as pairs. In the case of a pair with contrasting traits, the ‘dominant’ one appears in the hybrid offspring, although the ‘recessive’ one is also present.

    1.6 Gene and genetics

    Ironically, Mendel’s findings were not recognized during his lifetime. It was only at the turn of the century that his work was rediscovered and a spurt of activities in relation to the laws of heredity ensued thereafter. In 1909, Danish botanist Wilhelm Johanssen coined the term gene as a physical and functional unit of heredity. Earlier, in 1905, British geneticist William Bateson had introduced the term genetics.

    Now, the question was where in the cell the genes are located. The cell nucleus and the chromosome it contained were already known by then. Between 1825 and 1838, the nucleus was reported by three investigators including Robert Brown, who is credited for coining the term nucleus. Subsequently, German anatomist Walther Flemming, who is said to be the founder of the science of cytogenetics, was the first to observe and systematically describe the movement of chromosomes in the cell nucleus during normal cell division. In 1915, American geneticist Thomas Morgan and his students asserted that genes are the fundamental units of heredity. Their research confirmed that specific genes are found on specific chromosomes and that genes are indeed physical objects. The chromosome theory of inheritance emerged.

    1.7 Nature of physical objects in the cell

    Once it was established that genes are physical objects, the next step, as expected, was to investigate the physical (and chemical) nature of these objects. Major advances were already made in the investigation of another kind of important physical objects of the cell—the proteins.

    In 1789, French chemist Antoine Fourcroy had recognized several distinct varieties of proteins (though the term was not used then) from animal sources. These were albumin, fibrin, gelatin, and gluten. Several years later, in 1837, Dutch chemist Geradus Johannes Mulder determined the elemental composition of many of these molecules. The term protein was subsequently used by Mulder’s associate Jacob Berzelius to describe these molecules. The name was derived from the Greek word πρωτɛιοζ which means primary, in the lead, or standing in front.

    One class of proteins are the enzymes which catalyze the biological processes in living organisms. The first enzyme to be discovered was diastase (now called amylase). It was extracted from malt solution at a French sugar factory by Anselme Payne in 1833. The term enzyme was coined in 1878 by German physiologist Wilhelm Kühne. In 1897, Eduard Buchner, a German chemist and zymologist, fermented sugar with yeast extracts in the absence of live organisms.

    Nucleic acid, on the other hand, was discovered by a Swiss physician Friedrich Mischer. Working in the laboratory of biochemist Felix Hoppe-Seyer at Tübingen, Mischer initially intended to study proteins in leucocytes (blood cells containing nuclei). However, in his experiments he noticed a precipitate of an unknown substance. On further examination, the substance was found to be neither a protein nor a lipid. Unlike proteins, it contained a large amount of phosphorous. Since the substance was from the cells’ nuclei, it was named nuclein.

    Later, Albrecht Kossel, another scientist in Hoppe-Seyer’s laboratory, found that nuclein consisted of four bases and sugar molecules. Kossel provided the present chemical name nucleic acid. In 1909, a Russian-born American scientist, Phoebus Levine, isolated nucleotides, the basic building blocks of ribonucleic acid (RNA).

    1.8 DNA as the genetic material

    All these advances notwithstanding, till the late 1920s, the question regarding the nature of the genetic material remained unanswered. The scenario started changing in 1928 when British bacteriologist Fred Griffith carried out an experiment on the pathogenicity of Streptococcus pneumonia. Though not conclusive, the experiment did lay the foundation for later discovery that DNA is the genetic material. The results showed that apparently something in the cell debris of a virulent strain of Streptococcus had transformed an avirulent strain to become virulent. This something was called the transforming principle; it remained unclear what the transforming principle was—RNA, DNA, protein, or a polysaccharide.

    The conclusive evidence was provided 16 years later, in 1944, by American microbiologists Oswald Avery, Colin MacLeod, and Maclyn McCarty. They established that the active genetic principle was DNA since its transforming activity could be destroyed by deoxyribonuclease, an enzyme that specifically degrades DNA.

    A confirmation of the conclusion made by Avery and his colleagues came in 1952 from two scientists, Alfred Hershey and Martha Chase, who were working at Cold Spring Harbor Laboratory. The protein coat and DNA core of a bacteriophage were labeled with ³⁵S and ³²P, respectively. By infecting a bacterial culture with the radiolabeled phage, they showed that the parental DNA, and not the parental protein, was present in the progeny phage.

    1.9 Biology turns molecular—natural science becomes unified

    Undoubtedly, with the discovery of the two most important macromolecules of the living cell—DNA, that carries the blueprint of life, and protein, that executes the plan—biology had turned molecular. In fact, as early as in 1938, Warren Weaver, who was the director of the Natural Sciences section of the Rockefeller Foundation at that time, introduced the term molecular biology. The cellular processes were now required to be explained in terms of molecular interactions. It became ever more evident that the living system conforms to the laws of physics and chemistry.

    Eminent geneticist Hermann Mueller recognized the similarity between the contemporary developments in physics and genetics. In 1936, he even made a fervent appeal to the physicists and chemists to join forces with him and his colleagues in unraveling the fundamental properties of genes and their actions. Eight years later, in a book entitled What is Life, Erwin Schrödinger, one of the founders of quantum mechanics, made a similar plea and expressed his thoughts, many of which were similar to those of Mueller.

    Physicists indeed joined forces with biologists with the goal of understanding how the two disciplines complemented each other. As Alexander Kitaigorodsky, a renowned physicist and passionate popularizer of science, wrote (1971)—Physics is gaining a leading position in biology. This is not to be understood that physics is conquering biology and shoving biologists to the background. Simply the science we used to call biology is evolving into physics, thereby confirming the view that natural science is passing through a period of reconstruction on a unified foundation. Several X-ray crystallographers as well as structural chemists, such as Linus Pauling, dedicated their knowledge to the investigation of macromolecular structure.

    1.10 Deeper into the structure of matter

    We have already seen that the contribution of physics-based techniques to biology had begun with the development of the microscope. It marked the beginning of the studies of cell structure—structural cell biology. Thereafter, improvements in microscopy have facilitated ever more revelations of the cellular and subcellular structures.

    Nevertheless, visible light-dependent microscopy had its limitations. As the resolution was inversely proportional to the wavelength of the observing light, the microscope had reached the theoretical limit of its resolving power. The problem was overcome by the introduction of the electron microscope. Here, visible light waves are replaced by electron waves which are bent and focused by electromagnetic lenses. The first electron microscopes were manufactured in the 1930s. However, traditional electron microscopes were not able to determine the precise structures of macromolecules such as proteins and nucleic acids.

    Molecules, including biomolecules, are made up of atoms and bonds with dimensions around 0.1 nm. Three different methods can provide structural information at this resolution—X-ray crystallography, nuclear magnetic resonance (NMR) spectroscopy, and electron crystallography.

    X-rays were discovered by Wilhelm Conrad Röntgen in Germany in the year 1895. Seventeen years later, Max von Laue, speculating that the wavelength of X-rays might be comparable to the interatomic distances, observed the diffraction pattern from a blue crystal of copper sulfate. Then, in 1913, the father-and-son team of Henry Bragg and Lawrence Bragg determined the structure of crystalline common salt and initiated the field of X-ray crystallography. As we shall see below, X-ray crystallography has become the most widely used technique in the structural analysis of biological molecules.

    The phenomenon of nuclear magnetic resonance was first observed in 1946 by the physicists Felix Bloch of Stanford University and Edward Purcell of Harvard University independent of each other. In 1950, Warren Proctor and Fu Chun Yu discovered that the two nitrogen atoms in NH4NO3 produced two different NMR signals—a phenomenon that became known as chemical shift. This discovery, together with the contributions of physicist Albert Overhauser and many others, has made NMR spectroscopy the second most useful tool in the field of structural biology.

    Electron crystallography is based on electron diffraction phenomenon. In 1968, David DeRosier and Aaron Klug laid the foundation of electron crystallography by demonstrating that electron microscopic images of a two-dimensional crystal generated at different angles to the electron beam could be combined with electron diffraction data to produce a three-dimensional structure. Since then, electron crystallography has been used to determine the structures of a relatively few but important proteins.

    Understandably, with each of the above technical advances, view of the cell became more distinct and enlarged than before. From the whole cell and its organelles, structural investigation could go down to the molecular level.

    1.11 Molecular biology endowed with structures

    By the early 1950s, DNA had become a prominent target for structural studies. Several investigators engaged themselves in obtaining X-ray diffraction photographs of DNA fibers. The most prominent of them were those by Rosalind Franklin at King’s College, London. The photographs she obtained were, according to John Desmond Bernal (affectionately called Sage), among the most beautiful X-ray photographs of any substance ever taken.

    At the same time, Francis Crick and James Watson were working together at the Cavendish Laboratory, University of Cambridge, to build a model of DNA. Based on the X-ray diffraction data of Rosalind Franklin and Maurice Wilkins (also at King’s College) and profound theoretical insight of Crick, they published the double helical structure of DNA in the journal Nature in 1953.

    It is interesting to note here that the proposal of the double helical structure of DNA, though helping to explain two vital dynamic processes of the living cell—DNA replication and DNA recombination—was verified at atomic detail by single-crystal diffraction much later, in the late 1970s. The crystal structures of all forms of the DNA double helix—A, B, and Z—were published during this period. On the other hand, the structure of a complete A-RNA turn was visualized only in the late 1980s.

    Around the same time at the Cavendish Laboratory, Max Perutz and John Kendrew were pursuing X-ray crystallographic studies of proteins since the 1940s. Perutz was working on horse hemoglobin and Kendrew on a protein four times smaller, sperm whale myoglobin. It was Kendrew who, in 1957, met with the first success which was, nevertheless, aided by Perutz’ solution to the phase problem. The structure of myoglobin was initially determined at 6 Å resolution, soon improved to 2 Å. In 1960, Perutz published the structure of hemoglobin determined at 5.5 Å and later improved to 2.8 Å. It will not be inappropriate to affirm here that with the discovery of the structures of these crucial biological macromolecules, first the DNA and then the two proteins, the age of molecular structural biology had begun.

    Following the successes of Perutz and Kendrew, the structure of seven additional proteins were published in the 1960s; hen egg white lysozyme was the first enzyme whose structure was known in 1965. Nevertheless, all the proteins including enzymes, whose crystal structures had been determined till the late 1970s, were water-soluble. In contrast, membrane-located proteins, being insoluble, were more difficult to crystallize. The first membrane protein to be crystallized in 1980 was Halobacterium halobium rhodopsin, whereas the photosynthetic reaction center (PSRC) from Rhodoppseudomonas viridis became the first transmembrane protein to have its three-dimensional structure determined by X-ray crystallography in 1984.

    1.12 Structural complex(ity) disentangled

    From single polypeptide chains, X-ray crystallography progressed towards multi-subunit structures. In 1989, the three-dimensional structure of Escherichia coli RNA polymerase core, a five-subunit enzyme, was determined by electron crystallography. This was closely followed by the determination of the three-dimensional structure of yeast RNA polymerase II, an enzyme more complex than the bacterial polymerase, at 16 Å resolution. Eventually, in 2001, Roger Kornberg and his colleagues at Stanford University published the three-dimensional structure of a 10-subunit variant of the yeast enzyme (out of the complete set of 12 distinct polypeptides), determined by X-ray crystallography at 2.8 Å resolution. The publication enormously facilitated a better understanding of the multiple steps of RNA polymerase II-mediated transcription in eukaryotes.

    Another remarkable feat in macromolecular structure analysis was the determination of the structure of the ribosome—a huge molecular complex that translates genetic messages in the living cell. Ribosomes, prokaryotic or eukaryotic, consists of two subunits—small and large. Each subunit contains one to three RNAs and several proteins. Initially, the crystals of complete ribosomes were refractory to X-ray diffraction studies. Therefore, individual subunits were crystallized and their structures determined separately. The structure of a bacterial ribosomal subunit was first known in 1987. The eukaryotic subunits, which were more complex than their bacterial counterparts, had to wait till the turn of the century to have their structures resolved.

    1.13 Molecular structural biology confronts human disease

    In the last few decades, molecular structural biology has made a profound impact on the understanding of human disease and discovery of its remedy. Preliminary efforts in this direction had started with the publication of hemoglobin structure. Later in the 1980s, a large number of pharmaceutical companies became interested in utilizing structural data to design therapeutic molecules that would target specific proteins or nucleic acids. Among others were Agouron Pharmaceuticals in California, USA, and Molecular Discovery in UK.

    A shining example of the rational drug design effort in the 1990s has been the development of drugs for the treatment of human immunodeficiency virus (HIV) infection. Work on the inhibitors of HIV aspartic protease, whose structure was already available, led to Food and Drug Administration’s (FDA) endorsement of four successful drugs. Additional protease inhibitors were developed later and, eventually, a fatal disease became a manageable infection.

    1.14 From gene to genome

    It is no surprise that genetics, which studies individual genes and their roles in inheritance, has its limitations. The study of individual genes, one or only a few at a time, gives partial information on most of the metabolic processes and interaction networks within and across the cells of an organism. So, it became both necessary and important to focus on the entire genome, which is the complete set of DNA (including genes) of an organism. The science of genomics emerged towards the end of last century.

    As the focus shifted (or rather expanded) from gene to genome, a global effort to determine the genome sequences of various organisms—from bacteria to human—was underway. In 1995, the genome sequence of the bacterium Haemophilus influenzae was published and, by 2007, sequence of the entire human genome was declared to be finished.

    Concurrently, the number of protein sequences, mostly derived from genome sequences by bioinformatic means, kept growing with unimaginable rapidity. Advances in structure determination were relatively slower. As a result, the gap between genomic and structural information was widening. To address the issue, several structural genomics (SG) initiatives were created in America, Europe, and Asia at the beginning of the 21st century. The aim of structural genomics is to create a representative set of three-dimensional structures of all macromolecules found in nature. With this objective, the SG centers have put special efforts into the development of high-throughput methodologies for faster and more accurate determination of protein structures.

    1.15 In lieu of a conclusion

    Thus, it appears that the genomic and postgenomic projects are rapidly moving towards providing sequence as well as structural information on the entire set of molecular components present in an organism. As a welcome consequence, molecular biology is no longer restricted to the studies of single (or even a few) macromolecules. It is becoming ever more competent to understand and predict the behavior of biological systems based on a set of molecules involved and the relationships between them.

    References and Further Reading

    Bernal J.D. Science in History. London: Faber & Faber; 2010.

    Brooks-Bartlett J.C., Garman E.F. The Nobel science: one hundred years of crystallography. Interdiscip. Sci. Rev.. 2015;40(3):244–264.

    Campbell I.D. The Croonian lecture 2006 structure of the living cell. Philos. Trans. R. Soc. B. 2008;363:2379–2391.

    Dahm R. Friedrich Miescher and the discovery of DNA. Dev. Biol.. 2005;278:274–288.

    Grabowski M., et al. The impact of structural genomics: the first quindecinnial. J. Struct. Funct. Genom.. 2016;17(1):1–16.

    Harrison S.C. Whither structural biology. Nat. Struct. Mol. Biol.. 2004;11(1):12–15.

    Jaskolski M., Dauter Z., Wlodawer A. A brief history of macromolecular crystallography, illustrated by a family tree and its Nobel fruits. FEBS J.. 2014;281:3985–4009.

    Kitaigorodsky A. I Am a Physicist. Moscow: MIR Publishers; 1971.

    Masters B.R. History of the optical microscope in cell biology and medicine. In: Encyclopedia of Life Sciences (ELS). Chichester: John Wiley & Sons; 2008:doi:10.1002/9780470015902.a0003082.

    Mazzarello P. A unifying concept: the history of cell theory. Nat. Cell Biol.. 1999;1:E13–E15.

    Schrödinger E. What is Life. Cambridge, UK: Cambridge University Press; 1967.

    Shi Y. A glimpse of structural biology through X-ray crystallography. Cell. 2014;159:995–1014.

    Chapter 2

    Mathematical tools

    Abstract

    Scientific investigation of the structure and dynamics of a physical object or system requires (a) the measurement of different physical quantities and (b) appropriate mathematical tools. This chapter presents a brief description of the standards of measurement followed by an overview of the mathematical concepts that are important for quantitative analyses of molecular structure and dynamics. Physical systems (including biological systems) and their dynamics are quantitatively described in terms of observables or entities which can be measured. The entities are denoted by variables and functions are defined to reveal the interdependence of different variables. The function can be algebraic, trigonometric, etc. and can sometimes be represented by an infinite series. Distinction is made between a scalar quantity with only magnitude and a vector having both magnitude and direction. Changes in the function with respect to the variable/variables on which it depends are discussed in the section on

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