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© 2016 Dr. Volker Egelhofer
Herstellung und Verlag:
BoD – Books on Demand GmbH, Norderstedt
ISBN: 978-3-7412-5505-2
Bioinformatics links content from the life sciences with mathematical concepts from computer science and is significantly involved in the current advances in the molecular biology and medicine. Bioinformatics is not as misleadingly claimed in some publications a sub-field of systems biology, rather the systems biology is an important application field of bioinformatics. Further areas of application are the structural biology, the pharmaceutical and biotechnological research, the Genome-, Proteome-, Transcriptome- and Metabolom Analysis. A core field of Bioinformatics comprises the development of software for the storage, evaluation and analysis of scientific data. However, the main task of bioinformatics is in the development of reliable algorithms for predicting biological functions of large biological datasets, as well as the development of algorithms for the simulation of biological processes based on these data. Bioinformatics is an independent science, because only with the help of sophisticated bioinformatics algorithms it will be possible to generate usable knowledge for the prediction of biological functions based on such large amounts of data. The long-term goal of this discipline is the computer-aided simulation of all known life processes of a human, animal or plant cell. As a result, the medium-term objective for Bioinformaticians will be gathering, storage and analysis of information about crucial biochemical processes of living cells. In this context, the development of biological databases plays an important role. Another medium-term goal will be analysis of the interactions of individual cells with each other, as well as with their environment. Moreover, there are in particular molecules and biochemical reactions of note, whose function is altered based on gene- or protein modifications. In this context, bioinformatics algorithms will be used to determine the function of individual genes and proteins as well as to identify the regulatory components of the cells. A prerequisite for the understanding of the content as presented in the book requires a basic knowledge of biology, chemistry and biochemistry in particular. For this reason, at the beginning of this book the required basics of the mentioned disciplines, important methods in molecular biology, essential working techniques and the basics of probability and statistics will be introduced.
The book is addressed to graduate and undergraduate students in the life sciences and information technology, as well as to advanced researchers in this fields, who want to acquire basic knowledge in the field of Bioinformatics. It serves as an entry point to the discipline as well as for deepening existing knowledge.
Volker Egelhofer studied biotechnology, sinology, computer science and biochemistry. He obtained the degree Master of Engineering in biotechnology at the University of Applied Science Berlin and the degree Master of Science at the free University of Berlin. He performed his doctoral thesis in the field of theoretical biochemistry and bioinformatics at the Max Planck Institute for molecular genetics and acquired the degree of a doctor of science from the free University of Berlin in 2002. He held lectures and seminars in the field of bioinformatics at the University of Cologne, Technical University of Brunswick and the University of Vienna. He has more than 15 years' experience in the development of Bioinformatics algorithms and is the author of a number of scientific publications.
Chemistry is a branch of science. In following chapters, the basics of general and physical chemistry are taught, whose understanding is a prerequisite for a successful entry into the bioinformatics.
Atoms are the basic building blocks of matter. Every atom is composed of a nucleus and a nuclear envelope. The nucleus consists of protons and neutrons. The electrons move around at high speed in a certain distance to the nucleus. Protons have a positive and electrons a negative charge. Neutrons, on the other hand are not charged. The nucleus of an atom is very small in relation to its volume, but represents the total positive charge as well, due to the negligible mass of the electrons, the total mass of the atom. The electrostatic repulsion of the protons due to their positive charge is prevented by the nuclear strong force.
An uncharged atom is made up of the same number of protons and electrons. The number of electrons in an atom is the same as the number of protons. The mass number is the sum of the number of protons and the number of neutrons. Only its atomic number is decisive for the chemical properties of an atom. Isotopes are atoms with the same atomic number but different mass number. The different mass relies on the different number of neutrons in the nucleus of the isotopes. The most natural elements consist of mixtures of isotopes. Carbon for example is a mixture of the isotopes C12 (> 98.9%), C13 (~ 1%) and C14 (traces). Exceptions are sodium or fluorine. The distribution of the isotopes and the resulting isotope patterns play an important role especially in the mass spectrometry (see Chapter 27).
The term electronegativity ' describes the ability of a nuclear core to attract its electrons. The absolute values are not of importance, only the relative values are important for comparing the ability of atoms to bind electrons to the nucleus. The element with the highest electronegativity (4.0) is fluorine followed by oxygen (3.5), nitrogen (3.0). Hydrogen has only value of 2.1.
The spectrum of electromagnetic radiation range from low-energy, long-wave radio waves to high-energy, short-wave gamma rays (see table 1). Electromagnetic radiation can be represented as both wave and particle flow. Electromagnetic waves travel at the speed of light c in a vacuum. Two waves can superpose to form a wave of greater or lower amplitude. The electromagnetic radiation consists of an oscillating electric field that is perpendicular to an oscillating magnetic field. The relation between the wavelength λ and the frequency ν is described by the following formula
(1.1)
In quantum theory, the electromagnetic radiation is described as a stream of Photons. The energy of a photon is represented by following formula:
(1.2)
with the Planck's action quantum h =6,6261*10-34Js.
Thus, the higher the frequency of radiation, the higher the energy of its photons (for example the energy of UV light is higher than the energy of visible - or infrared light).
Table 1: Ranges of electromagnetic radiation
Radiation | Wavelength | Frequency [Hz] |
Radio waves | >30cm | 3KHz - 300 MHz |
Micro waves | 1mm - 30cm | 300MHz - 30 GHz |
Infrared | 800nm - 1mm | ~1-380THz |
Visible light | 400-800nm | ~380-800THz |
Ultraviolet | 3-380nm | ~800THz-300PHz |
X-rays | ~50pm- ~10nm | 30-300EHz |
Gamma rays | ~1- 50pm | 30-300EHz |
The Bohr model of the atom from 1903 paved the way for the understanding of the structure of the nuclear envelope. The model is based on the assumption that the electrons of an atom orbiting its nucleus only on certain tracks, similar to the orbit of the planets around the Sun (see fig. 1.1A). The energy of an electron is proportional to its distance from the nucleus. The circular paths on which the electrons move around the nucleus are called electron shell. These shells are named with letters (K, L, M...) or with numbers (n = 1,2, 3…). The electrons, which move on the K-Shell (n = 1) have the lowest energy level, they are in the so-called ground state. Electrons, which are located on outer paths have higher energy levels, they are in a so-called excited state. Within in the Bohr model, the energy of an electron as a function of its orbit (n) is calculated with the following equation:
(1.3)
The amount of energy, that must be supplied, in order to transfer electrons from the inner paths to outer trajectories, is equal to the amount of energy, that would have to be spent to move the electron against the attraction of the positive protons in the atomic nucleus. Conversely, a certain amount of energy is released when electrons from outer paths fall back to inner tracks. In this case, the energy is emitted in the form of a light quantum (Photon), whose frequency is calculated according to Equation 1.2. The frequency of the released energy can be calculated with equation 1.4:
(1.4)
However, the Atomic model of Bohr is in some points contrary to real measurements. For example, is it not possible to represent an electron as a wave. However, a crucial contradiction is the non-fulfilment of the Heisenberg uncertainty principle.
The Heisenberg uncertainty principle states, that the position of a particle and its velocity cannot be determined simultaneously with a sufficient accuracy.
The relation between the uncertainty of the location Δx and the uncertainty of the velocity, respective of the pulse Δp is described by equation 1.5:
(1.5)
The idea of defined, discrete circuits on which the electrons move around their nucleus, is in contrast to the Heisenberg uncertainty principle, due to the fact that for an exact calculation of the electron circuits, it is a prerequisite to know the exact location and the exact velocity of the electrons at a given time.
The consideration of the light as a beam of particles (the photons) is fundamental in quantum mechanics. This approach was first introduced by Max Planck and Albert Einstein at the beginning of the 20Jh. According to Louis de Broglie, a photon can be regarded as a wave. The mass of a photon can be calculated with Einstein's equation 1.6
(1.6)
The energy of a photon is calculated using equation 1.2, in combination with equation 1.1 to get equation 1.7:
(1.7)
and by inserting eq.1.7 in eq. 1.6 is finally obtained eq.1.8:
(1.8)
However, not only the wavelength of a photon, but also those other flying particles - such as electrons - can be calculated. For that purpose, the speed of light c is replaced by the speed of the particle v (see eq. 1.9):
(1.9)
In the quantum mechanical model, the electrons will not be arranged on defined circuits, rather their whereabouts, called Orbitals, are calculated based on probability densities. Orbitals are described mathematically as wave functions. The wave functions of an electron in an atom can be calculated using the Schrödinger equation (eq. 1.10).
(1.10)
The Schrödinger equation is a differential equation and has therefore infinitely many solutions. But only the physical solutions, that meet certain conditions are reasonable. Each solution is equivalent to a specific value of the energy E. The square of the absolute value of the Wave function |ψ|2 indicates the probability density of the particle (after Max Born).
A schematic representation of the electron orbitals is shown in Figure 1B. There are three so called quantum numbers to describe the orbitals. The principal quantum number n = 1, 2, 3...is roughly equivalent to the electron shell in the Bohr 's atom model. Each of these main shells can be divided into sub shells. The number of sub shells corresponds to the number of the principal quantum number, thus, if n = 1, there is exactly one subshell, and so on. Each sub shell is named by a secondary quantum number l = 1, 2, 3...or with a letter l = s, p, d, f. The number of orbitals in a subshell is calculated as 2l + 1 (see table 2). Orbitals can be associated with two nuclei and therefore form chemical bonds.
Table 2: The number of orbitals of different Electron shells.
n | l | Sub shell | Orbital (2l+1) |
1 | 0 | 1s | 1 |
2 | 0 | 2s | 1 |
1 | 2p | 3 | |
3 | 0 | 3s | 1 |
1 | 3p | 3 | |
2 | 3d | 5 |
Molecules consists of one or more atoms of the same or of different type held together by chemical bonds. The chemical linkage can be different. There are covalent bonds, ionic bonds, hydrogen bonds and bonds due to London forces (see fig. 2.1).
The covalent bond is formed between two atoms when a pair of electrons, one of each atom, sharing a common orbital. Together they form a so-called electron pair (see fig. 2.1A). This electron pair (covalent) bond is a strong chemical connection, therefore a relatively large amount of energy is required to break the bond between them. The covalent bond is predominantly build between nonmetals atoms. There are single -, double -, and triple covalent bonds.
The energy of a covalent bond is proportional to the number of participating electron pairs. The bond between two electrons within molecules is symbolized by lines drawn between them, representing a saturated valence for each atom. The electrons of some molecules (e.g. 'benzene') form a kind of resonance structure, which achieves a stabilization by a permanent relocation of the participating electrons. This effect is called mesomerism or resonance. In such molecules, the exact positions of the double and single bonds between the participating atoms is not exact predictable. For this reason, the bonds between the atoms of such molecules will be drawn with a dotted line instead of as usual with continues line. In ring structures the resonance is indicated by a circle. The total number of possible bonds, which can be established by a single atom at once, depends on its number of free single occupied orbitals. For example, hydrogen has only one single occupied s-orbital and therefore can only form one bond. On the contrary, nitrogen has three single occupied p-orbitals and may form up to three bonds. According to Lewis [1] the atoms attempt to achieve the so-called noble gas configuration, by bonding in such a way that each atom has eight electrons in its valence shell. In the case of hydrogen, the goal is to achieve the helium configuration (2 electron configuration), for all other atoms (> 2 electrons) it is the aim of achieving the 8-Elektronenkonfiguartion (octet rule) of the other noble gases. Atoms can also form multiple bonds (see table 3).
Table 3: Examples of multiple bonds between atoms
Bond | Example | Description |
Single | C-H | Between carbon C and hydrogen H |
Double | C=C | Between 2 carbon atoms |
Double | N≡N | Between 2 nitrogen atoms N |
The Ionic bond is formed between metals and non-metals and their bond strength is approximately in the range of the covalent bonds. This type of bond is in contrast to the covalent bonds, undirected and has a longer range. It is usually formed between atoms in salts. Figure 2.1B shows schematically the binding between a sodium (Na+)-Ion and a potassium (K-)-Ion. The force behind the Ionic bonds are intramolecular electrostatic interactions. The strength of electrostatic interactions and thus the strength of an ionic bond can be described by Coulomb's law (see eq. 2.1):
(2.1)
The strength of the electrostatic attraction between two ions and therefore their bond strength is directly proportional to the product of their charges and inversely proportional to the square of their distance.
Beside intramolecular interactions, as the already mentioned ion bonds, there are also Intermolecular interactions, i.e. electrostatic bonds between two different molecules.
An electric dipole can build within a molecule that consists of atoms of different electronegativity. The dipole takes effect, if the atom with the higher electronegativity pulls the shared binding electrons more closely to its nucleus. As a result of the electron movement a locally negative charge is built at the side of the atom with the higher electronegativity and in contrast due to the loss of its electron, remains on the other side a positive charge. A distinction is made between temporary - and permanent dipoles. Temporary dipoles may occur through short-term charge shifting, induced from other molecules (see more details on that in the Chapter: London interactions). In addition to the requirements for building temporary dipole's, the building of a permanent dipole needs a specific geometrical arrangement of the involved atoms within the molecule. That is why, the carbon dioxide molecule (CO2) is not a permanent dipole, because its atoms are arranged in a single row respective in single layer, which in turn results in a charge neutralization (see fig. 2.1 C, right). On the contrary, a typical example of a permanent dipole, is H2O. The oxygen in the water molecule pulls the shared electron pair closely to its nucleus, inducing therefore a negative partial charge on its side, as well as positive partial charges on the hydrogen atoms in the oxygen atom (see fig. 2.1 C, left). Due to the asymmetric arrangement of the involved atoms, the charge shifting within the molecule remains permanently.
Hydrogen bonds are the strongest intramolecular electrostatic interactions. As the name implies, the hydrogen atom is bridging between two strong electronegative atoms. Roughly speaking, both electronegative atoms involved in the bond are trying to pull the hydrogen atom over to their side. Generally, formulated: δ-X - Hδ+ .... δ-Y, where X and Y symbolize one of the following elements: Fluorine (F), oxygen (O), or nitrogen (N) (see fig. 2.1D). The hydrogen bonds in the Helix structure of proteins (see Chapter 9.3) are mainly responsible for the formation of the proteins 3-dimensional structure (see Chapter 9.3).
London dispersion forces are very weak, when compared to the above described bonds. They are induced by a molecule, which forms a temporary dipole (see fig. 2.1E, step 1). Now, this temporary dipole is able to polarize other nearby unipolar molecules, which induce the formation of a dipole in these molecules too (see fig. 2.1E, step 2). As a result, both molecules attract each other (see fig. 2.1E, step 3). Now, if within the initial dipole molecule, the direction of the charge is changed, the induced molecule will follow with a corresponding turnaround. Both molecules are more or less connected, in a way, that their interaction never reach zero, but leads to the London dispersion forces. The strength of these forces depends on the polarizing capability of both molecules.
Acids, also called Proton donors are substances which can donate protons (H+) to their environment. Bases take up protons, and are therefore called Proton acceptors. In general terms, an acid act as electron-pair acceptor and a base as an electron-pair donor. A base has a lone electron-pair, which can be used by an acid particle to build a pair of electrons, to establish a covalent bond.
H2O dissociates in H+- und OH-Ions. The equilibrium constant KW is the ion product of water and is at 25° C: 10-14 mol2/l2 (eq. 3.1).
(3.1)
The pH-value is calculated using EQ. 3.2:
(3.2)
An acidic solution has a pH-Value between >=0 and <7, a basic solution between >7 and <=14. At pH=7 is the solution neutral.
In contrast to strong acids such as hydrochloric acid (HCl) and sulfuric acid (H2SO4), weak acids such as the acetic acid (CH3CH2OH) or amino acids do not dissociate completely in aqueous solutions. For the dissociation process of weak acids in dilute aqueous solutions, the amount of water is constant. Therefore, the dissociation constant Ks is calculated by means of the law of mass action. In equation 3.3, the calculation of Ks is exemplarily shown for acetic acid.
(3.3)
The pKS-Value can be calculated with the following formula:
(3.4)
In case of weak acids, the dissociation constant KS is small, therefore the pH-Value can be calculated with the Equation 3.5:
(3.5)
Thus, the pKS-Value is a measure of the strength of a weak acid, the smaller this value, the stronger is the corresponding acid. Analog to acids, a KB-Value and pKB-Value can be calculated for bases.
During a chemical reaction, the reactants (educts, chemical precursor) are converted into the products. A chemical reaction can be written as a reaction equation. The reactants will be written on the left side and the products on the right side, between them is an arrow, indicating the direction of the chemical reaction (eq. 4.1).
(4.1)
A double arrow (↔) indicates, that the chemical reaction is at equilibrium. A reaction equation must be stoichiometrically correct, i.e. an equal number of atoms must be located on both sides of the reaction arrow. Chemical reactions follow the laws of thermodynamics. Every chemical reaction can be in principle proceed in both directions, but mostly the balance is on the side of the reactants, therefore no product formation takes place. By calculating the change in the Gibbs Free Energy (Gibbs enthalpy) ΔG (eq. 4.2) of a chemical reaction, it can be determined, whether the reaction will proceed under the given physical conditions (pressure, temperature) and at given concentrations of the reactants and the products, without the presence of a catalyst (see table 4).
(4.2)
ΔH stands for the change of enthalpy, i.e. the increase or decrease of the internal energy of the system and ΔS stands for the change in the entropy of the system. The enthalpy H is defined by the following equation 4.3:
(4.3)
pV is the volume work, i.e., the energy that is needed, to change the size of a given volume on a different size. U is the complete internal energy in the entire thermodynamic system. The change in the internal energy ΔU is the product of heat and work and is constant in a closed system (first law of ther-30 modynamics). The entropy S describes the state of organization of all components in the system and can be the calculated with the Boltzmann equation (eq. 4.4).
(4.4)
kB (1,381•10-23 J/K) is the Boltzmann constant;
Ω is the number of different possibilities, that the atoms can arrange themselves in a closed system in space. The entropy increase with the existing number of possibilities for arranging the atoms in the available space, or in other words the higher the disorder in the system, the higher is the entropy.
Table 4: Gibbs Free Energy (Gibbs enthalpy) ΔG
ΔG | Reaction | Description |
<0 | exogenous | Proceed without any additional energy or catalysts. |
=0 | equilibrium | Rate of return and reverse reaction is equal |
>0 | endogen | Requires energy to proceed from outside (E.g. heat), or catalysts. |
A chemical reaction is exogenous, if the Gibbs free energy ΔG decreases during the reaction. The decline of ΔG is initialized by change of entropy of ΔS, an increase in temperature and/ or a change in enthalpy ΔH.
Biology, besides biochemistry is one of the classic disciplines in life sciences. For the computational biologists are especially the subareas of cell biology, microbiology and molecular biology of relevance, however, the latter more represents a branch of biochemistry. In cell biology, the cell as the smallest living unit of organisms is in the focus of interest. There are prokaryotic cells (Procyten) and eukaryotic cells (Eucyten). Living organisms, which consist of prokaryotic cells, called prokaryotes, and those that are built up from eukaryotic cells, called eukaryotes. Prokaryotes as well as eukaryotes may occur as single cells (bacterial cells respective yeast cells), as well as multi-cellular organisms. The human body consists of several trillions of cells.
The prokaryotes are divided in Bacteria and Archaea. Their cells consist of cytoplasm, which is surrounded by a single cell membrane. All metabolic reactions of the cell occur in the cytoplasm. The DNA of a prokaryotic cell is not surrounded by a membrane, rather the DNA is concentrated in a central area of the cell called the nucleoid.
Examples of eukaryotes are animals, humans, fungi and plants. Eukaryotic cells like the prokaryotic cells consist of cytoplasm with a surrounding cell membrane. However, their DNA is not free in the cytoplasm. The eukaryotic DNA is wounded to proteins called histones. The histones are part of the chromosomes in the cell nucleus (nucleus) of the cell. In addition to the nucleus, the eukaryotic cell contains several characteristic cell compartments (see table 5). The usage of compartments is advantageous, because within a compartment closely related metabolic steps can efficiently be carry out, without bridging long diffusion paths through the cytosol. Plant cells have some additional peculiarities, like an extra cell wall and specific cell compartments like, chloroplasts, vacuoles and Glyoxysomes.
Table 5: Compartments of eukaryotic cells
Compartment | Membrane | Important functions and features |
Nucleus | double | DNA-Replication, Transcription |
Mitochondrion | double | Synthesis of ATP, contain their own DNA |
Peroxisomes | single | β-Oxidation |
Endoplasmic reticulum (ER) | single | Either: Lipid synthesis, intracellular Ca+ storage; Or: studded with ribosomes (protein biosynthesis) |
Golgi apparatus | single | sugar phosphorylation of proteins. Zell transport |
Lysosomes | single | Cellular digest |
Vacuoles 1 | single | Storage of nutrients, depot for waste |
Glyoxysomes 1 | single | Glyoxylat Cycle |
Chloroplasts1 | double | Photosynthesis |
1 Special compartments only occurring in plants.
The main differences between Prokaryotes and Eukaryotes are summarized in table 6.
Table 6: Different Compartments in prokaryotes and eukaryotes cells
Prokaryoten | Eukaryoten | |
DNA | Without surrounding mem- brane in the Cytosol | Within the nucleus |
Ribosomes | 70S1 | 80S1 |
Size | 10-100μm | 1-10μm |
1S stands for Svedberg, the unit for the sedimentation coefficient. This coefficient depends on the mass and shape of the sedimenting particles.
Alberts, B., Bray, D., Lewis, M., Raff, M., Roberts, K. and Watson, J.D. "The Cell" Garland Science; 3 edition, 1994.
Biochemistry is the study of the molecular relationships of the most important biological and chemical processes in organisms. Therefore, a basic understanding of Biochemistry is a prerequisite for successful entry into the bioinformatics. The first part of this chapter introduces the four main substance classes of biochemistry: proteins, carbohydrates, lipids, and nucleic acids. The second part of the book gives a brief overview of molecular genetics. In the final part the basic metabolic processes will be discussed and selected metabolic pathways will be presented.
Proteins are built from amino acids. They fulfil a variety of biological functions. As structural proteins they providing both stability and elasticity of the organism and in their functional role as enzymes, they speed up the most essential chemical reactions of the metabolism.
Amino acids are organic compounds, consisting of one or more carboxyl groups (-COOH) and containing least one amino group (-NH2). The covalent bonds of the so-called α-C-Atom of an amino acid with a carboxyl group as well as with and an amino group is characteristic for the protein building α-amino acids. The covalently-bound carbon atoms of the backbone (main chain) of an amino acid molecule are named with small Greek letters, starting with the α for the first C-Atom after the carboxyl group. The 2D-Structure of the 20 most important proteinogenic α-Amino acids are shown in Figure 3. These amino acids can be divided into different groups according to their different side chains.
In amino acid sequences, an individual amino acid is usually represented by its single letter code or less common by its three-letter code (see fig. 9.1). The side chain determines the chemical function of an amino acid. The side chains are grouped based on their functional groups. A distinction is made between polar groups like Sulfhydryl groups (-SH) or hydroxyl groups (-OH) with hydrophilic properties and longer non-polar carbon chains with hydrophobic properties. The hydrophobicity index [1] specifies the strength of the hydrophobicity (water displacement) of the side chains of an amino acids.
The more negative this value, the more hydrophilic (water-loving) is the side chain. Furthermore, a distinction can be made between basic side chains (NH, NH2) and acidic side chains (-COOH). The pH-Value of the surrounding solvent determines the chemical state of the functional groups. As a function of the pH of the solution for each functional group in an amino acid a specific dissociation constant KS for acidic (e.g. COOH) or KB for basic (e.g.-NH2) groups can be calculated. Thus, depending on the pH of the solvent a functional group are either neutral, negative, or positively charged. In acidic solutions most of the functional groups are protonated, whereas in basic solutions they are deprotonated. At a specific pH-Value, the total number of negative charges equals the total number of the positive charges, thus the net charge of the entire amino acid molecule is zero. This special pH-Value is called the isoelectric point (pI) of an amino acid. In Organisms, most amino acids can be synthesized from Ketoacids. Amino acids, that cannot be synthesized by enzymes and must be supplied by food called essential amino acids. For the adult people, the following amino acids are essential: Leucine, Isoleucine, Methionine, Lysine, Phenylalanine, Valine, Tryptophan and Threonine.
Amino acids chains, which are composed of at least 2 and not more than 100 amino acids called a peptide. A molecule composed of two connected amino acids is called a Dipeptide, of three connected amino acids a Tripeptide and so on. Peptides, which are composed of several amino acids also known as Oligopeptides. Longer chains of amino acids (> 100 amino acid molecules) are called Proteins. The bond between two amino acids is called a peptide bond, which is formed by a chemical condensation reaction, i.e. by dissociation of a water molecule. The cleavage of a peptide bond is in turn done through hydrolysis, i.e. the cleavage of the molecule is done by the Incorporation of a water molecule (see fig. 9.2A).