Physical Biochemistry Lecture Notes

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Principles of Physical Biochemistry CHAPTER Biological Macromolecules 1.1 GENERAL PRINCIPLES In physical biochemistry, we are interested in studying the physical properties of bi­ ological macromolecules, including proteins, RNA and DNA, and other biological polymers (or biopolymers). These physical properties provide a description of their structures at various levels, from the atomic level to large multisubunit assemblies. To measure these properties, the physical biochemist will study the interaction of molecules with different kinds of radiation, and their behavior in electric, magnetic, or centrifugal fields. This text emphasizes the basic principles that underlie these methodologies. In this introductory chapter, we briefly review some of the basic principles of structure and structural complexity found in biological macromolecules. Most read­ ers will have already learned about the structure of biological macromolecules in great detail from a course in general biochemistry. We take a different point of view; the discussion here focuses on familiarizing students with the quantitative aspects of structure. In addition, this discussion includes the symmetry found at nearly all lev­ els of macromolecular structure. This approach accomplishes two specific goals: to illustrate that the structures of macromolecules are very well defined and, in many ways, are highly regular (and therefore can be generated mathematically); and to in­ troduce the concepts of symmetry that help to simplify the study and determination of molecular structure, particularly by diffraction methods (Chapters 6 and 7). This discussion focuses primarily on the structures of proteins and nucleic acids, but the general principles presented apply to other macromolecules as well, including poly­ saccharides and membrane systems. 1 2 Biological Macromolecules Chapter 1 1.1.1 Macromolecu les As a basic review of molecular structure, perhaps the place to start is to ask the question, What is a molecule? Here, the definition of a biological molecule differs slightly from the definition learned in chemistry. In organic chemistry, a molecule consists of two or more atoms that are covalently bonded in speoific proportions ac­ cording to weight or stoichiometry, and with a unique geometry. Both stoichiometry (the chemical formula) and geometry (the chemical structure) are important. Dichloroethylene, for example, has the specific chemical formula C H C1 • This, 2 2 2 however, does not describe a unique molecule, but rather three different molecules. The geometry for one such molecule is defined by the arrangements of the chlorine atoms, as in cis-1,2-dichloroethylene (Figure 1.1). Now, the identity of the molecule is unambiguous. In biochemistry, a single molecule is considered to be a component that has well-defined stoichiometry and geometry, and is not readily dissociated. Thus, to a biochemist, a molecule may not necessarily have all the parts covalently bonded, but may be an assembly of noncovalently associated polymers. An obvious example of this is hemoglobin. This is considered to be a single molecule, but it consists of four distinct polypeptides, each with its own heme group for oxygen binding. One of these polypeptide-heme complexes is a subunit of the molecule. The heme groups are non covalently attached to the polypeptide of the subunit, and the subunits are noncovalently interacting with each other. The stoichiometry of the molecule can also be described by a chemical formula, but is more conveniently expressed as the Molecule Stoichiometry Geometry (Structure) CI Cl cis-l,2-Dichloroethylene '" / C=C H / '" H H"moglobin Figure 1.1 Examples of molecules in chemistry and macromolecules in biochemistry. The simple com­ pound cis-l,2-dichloroethylene is uniquely defined by the stoichiometry of its atomic components and the geometry of the atoms. Similarly, the structure of a biological macromolecule such as hemoglobin is defined by the proportions of the two subunits (the a and J3-polypeptide chains) and the geometry by the relative positions of the subunits in the functional complex. Section 1.1 General Principles 3 composition of monomer units. The stoichiometry of a protein therefore is its amino acid composition. The geometry of a biological molecule is again the unique linear and three-dimensional (3D) arrangements of these components. This is the structure of a biochemical molecule. A macromolecule is literally a large molecule. A biological macromolecule or biopolymer is typically defined as a large and complex molecule with biological function. We will take a chemical perspective when dealing with macromolecules, so, be judged in terms of the number of components (atoms, for this discussion, size will functional groups, monomers, and so on) incorporated into the macromolecule. Complexity generally refers to the organization of the three-dimensional structure of the molecule. We will treat size and structural complexity separately. What is considered large? It is very easy to distinguish between molecules at the two extremes of size. Small molecules are the diatomic to multiple-atom mole­ cules we encounter in organic chemistry. At the upper end of large molecules is the DNA of a human chromosome, which contains tens of billions of atoms in a single molecule. At what point do we decide to call something a macromolecule? Since these are biopolymers, their size can be defined by the terms used in polymer chem­ istry, that is, according to the number of sugar or amino acid or nucleic acid residues that polymerize to form a single molecule. Molecules composed of up to 25 residues are called oligomers, while polymers typically contain more than 25 residues. This is an arbitrary distinction, since some fully functional molecules, such as the DNA­ condensing J-protein of the virus G4, contain 24 residues. The structure of biological macromolecules is hierarchical, with distinct levels of structure (Figure 1.2). These represent increasing levels of complexity, and are de­ fined below. Monomers are the simple building blocks that, when polymerized, yield a macromolecule. These include sugars, amino acids, and nucleic acid residues of the polymers described above. Primary structure (abbreviated as 1°) is the linear arrangement (or sequence) of residues in the covalently linked polymer. Secondary structure (abbreviated as 2°) is the local regular structure of a macro­ molecule or specific regions of the molecule. These are the helical structures. Tertiary structure (abbreviated as 3°) describes the global 3D fold or topology of the molecule, relating the positions of each atom and residue in 3D space. For macromolecules with a single subunit, the functional tertiary structure is its native structure. Quaternary structure (abbreviated as 4°) is the spatial arrangement of mUltiple distinct polymers (or subunits) that form a functional complex. Not all levels of structure are required or represented in all biological macro­ molecules. Quaternary structure would obviously not be relevent to a protein such as myoglobin that consists of a single polypeptide. In general, however, all biological macromolecules require a level of structure up to and including 2°, and typically 3° 4 Biological Macromolecules Chapter 1 Figure 1.2 Hierarchical organization of macromolecular structure. The structures of macromolecules are orga­ nized starting with the simple monomers to form the sequence in the primary structure, which folds into the local n.;gular helices of secondary structure, the global tertiary structure, and the association of folded chains to form complexes in the quaternary structure. for biological function. The relationship between these levels of structure is often presented in sequential order as P, followed by 2°, which is followed by 3°, and fi­ nally 4° (if present). This sequential relationship is a convenient means of presenting the increasing complexity of macromolecular structure; however, it is not clear that this is how a molecule folds into its functional form. The most recent models for pro­ tein folding suggest that a less compact form of 3° (often called a molten globule state, see Section 4.4.3) must occur first in order to form the environment to stabilize helices (2°). One of the goals in physical biochemistry is to understand the rules that relate these levels of structural complexity. This is often presented as the problem of predicting 3D structure (2° to 3°) from the sequence (1°) of the building blocks. The problem of predicting the complete 3D structure of a protein from its polypeptide se­ quence is the protein-folding problem. We can define a similar folding problem for all classes of macromolecules. We will see how this hierarchical organization of structure applies to the structures of proteins and nucleic acids, but first we need to discuss some general principles that will be used throughout this chapter for describing molecular structure. It should be em­ phasized that we cannot directly see the structure of a molecule, but can only measure its properties. Thus, a picture of a molecule, such as that in Figure 1.2, is really only a model described by the types of atoms and the positions of the atoms in 3D space. This model is correct only when it conforms to the properties measured. Thus, methods for determining the structure of a molecule in physical biochemistry measure its interac­ tions with light, or with a magnetic or electric field, or against a gradient. In all cases, we must remember that these are models of the structure, and the figures of molecules presented in this book are nothing more than representations of atoms in 3D space. It is just as accurate (and often more useful) to represent the structure as a list of these atoms and their atomic coordinates (x, y, z) in a standard Cartesian axis system. Section 1.1 General Principles 5 1.1.2 Configuration and Conformation The arrangement of atoms or groups of atoms in a molecule is described by the terms configuration and conformation. These terms are not identical. The configura­ tion of a molecule defines the position of groups around one or more nonrotating or around chiral centers, defined as an atom having no plane or center of sym­ bonds metry. For example, the configuration of cis-l,2-dichloroethylene has the two chlo­ rine atoms on the same side of the nonrotating double bond (Figure 1.3). To change the configuration of a molecule, chemical bonds must be broken and remade. A con­ version from the cis- to trans-configuration of 1,2-dichloroethylene requires that we first break the carbon-carbon double bond, rotate the resulting single bond, then re­ make the double bond. In biological macromolecules, configuration is most impor­ tant in describing the stereochemistry of a chiral molecule. A simple chiral molecule Configuration Cl Cl Cl H '" / '" / C=C c=c / '" / '" H H H Cl cis-l ,2-Dichloroethylene trans-l ,2-Dichloroethylene Sp3 intermediate o 00_ f'0 HC . H;: /.' . HC + : HOC ___ H . I:\ .. H H ___ OH . C _ /0-'" : CH 0H 2 HO CH 0H CH20H 2 L-Glyceraldehyde Planar achiral intermediate D-Glyceraldehyde Conformation 9 Cl R,o .,R Cl : ''':'' /, C :­ X' 'R Cl Kf H H H H Eclipsed gauche Staggered anti Figure 1.3 Configuration and conformation both describe the geometry of a molecule. The configuration of a molecule can be changed only by breaking and remaking chemical bonds, as in the conversion of a cis­ double bond to one that is in the trans-configuration, or in converting from the L- to the D-stereoisomer of a chiral molecule. Conformations can be changed by simple rotations about a single bond. Biological Macromolecules Chapter 1 6 has four unique chemical groups arranged around a tetrahedral atom (usually a car­ bon atom with Sp3 hybridization). To change the configuration or chirality of this molecule, we must break one bond to form a planar achiral intermediate, and re­ form the bond on the opposite side of the plane. The resulting molecule is the stereoisomer or enantiomer of the starting structure. The stereoisomers of a mole­ cule, even though they are identical in chemical composition, are completely differ­ ent molecules with distinct properties, particularly their biological properties. Sugars that have more than one chiral center have more complex stereochemistry. The conformation of a molecule, on the other hand, describes the spatial arrangement of groups about one or more freely rotating bonds. For example, 1,2- dichloroethane, the saturated version of dichloroethylene, has no restrictions to ro­ tation about the chemical bonds to prevent the chlorine atoms from sitting on the same or opposite sides of the central carbon-carbon bond. These positions define the gauche and anti structural isomers, respectively. In addition, the conformation can be eclipsed or staggered, depending on whether the groups are aligned or mis­ aligned relative to each other on either side of the carbon-carbon bond. The confor­ mation of a molecule thus describes the structural isomers generated by rotations about single bonds (Figure 1.3). A molecule does not require any changes in chemi­ cal bonding to adopt a new conformation, but may acquire a new set of properties that are specific for that conformation. The stereochemistry of monomers. The monomer building blocks of bio­ chiral molecules, with only a few exceptions. There are logical macromolecules are many conventions for describing the stereochemistry of chiral molecules. The stere­ ochemistry of the building blocks in biochemistry has traditionally been assigned ac­ cording to their absolute configurations. This provides a consistent definition for the configuration of all monomers in a particular class of biopolymer. For example, the configurations of sugar, amino acid, and nucleic acid residues are assigned relative to the structures of L- and D-glyceraldehyde (Figure 1.4). In a standard projection for­ mula, the functional groups of D-glyceraldehyde rotate in a clockwise direction around the chiral carbon, starting at the aldehyde, and going to the hydroxyl, then the hydroxymethyl, and finally the hydrogen groups. The configuration of the build­ ing blocks are therefore assigned according to the arrangement of the analogous functional groups around their chiral centers. Since glyceraldehyde is a sugar, it is easy to see how the configurations of the carbohydrate building blocks in polysac­ charides are assigned directly from comparison to this structure. Similarly, the con­ figuration of the ribose and deoxyribose sugars of the nucleic acids can be assigned directly from glyceraldehyde. Biopolymers are typically constructed from only one enantiomeric form of the monomer building blocks. These are the L-amino acids in polypeptides and the D-sugars in polysaccharides and polynucleotides. For an amino acid such as alanine, the chiral center is the CO carbon directly adjacent to the carboxylic acid. The functional groups around the CO carbon are analogous but not identical to those around the chiral center of glyceraldehyde. The L-configuration of an amino acid has the carboxylic acid, the amino group, the a-hydrogen and the methyl side chain arranged around the C", carbon in a General Principles 7 Section 1.1 Figure 1.4 Absolute configuration of monomer f'0 f'0 building blocks. The stereochemistry of the monomers in biopolymers are assigned relative to L- and D­ IH\ IH glyceraldehyde. Carbohydrates and the sugars of HOC"""" H HC""""OH nucleic acids are assigned directly according to the \..:,J rotation starting at the carbonyl group. For amino CH 0H acids, the stereochemistry is defined according to 2 " H2 L-Glyceraldehyde D-Glyceraldehyde the rotation starting at the analogous carboxyl group. 0- ° 0- ° "f' "f' ( I\ +H3N C ........ H H C""""NH3+ \..,J CH CH 3 3 " J L-Alanine D-Alanine manner analogous to the aldehyde, hydroxyl, hydrogen, and hydroxymethyl groups in L-glyceraldehyde. Conformation of molecules. Unlike the configuration of a macromole­ cule, the number of possible conformations of a macromolecule can be enormous because of the large number of freely rotating bonds. It is thus extremely cumber­ some to describe the conformation of a macromolecule in terms of the alignment of each group using the gauche/anti and eclipsed/staggered distinctions. It is much more convenient and accurate to describe the torsion angle e about each freely ro­ tating bond. The torsion angle is the angle between two groups on either side of a freely rotating chemical bond. The convention for defining the torsion angle is to start with two nonhydrogen groups (A and D) in the staggered anti conformation with e = -180°. Looking down the bond to be rotated (as in Figure 1.5) with atom A closest to you, rotation of D about the B - C bond in a clockwise direction gives a positive rotation of the bond. Thus, the values for e are defined as 0° for the Torsion Angle Staggered Eclipsed Staggered i Figure 1.5 Torsion angles and dihedral angles (8). The rotation around a single bond is de­ Dihedral Angle scribed by the torsion angle of the four atoms e C B around the bond (A - B - C - D) and the D dihedral angle 8 relating the planes defined by atoms A-B-Cand byB-C-D. Biological Macromolecules Chapter 1 8 0 eclipsed gauche conformation to + 180 for the staggered anti conformation. Notice 0 that the start and end points «(J = ± 180 ) are identical. The angle between the two groups of atoms can also be defined by the dihedral angle. Mathematically, the dihedral angle is defined as the angle between two planes. Any three atoms about a freely rotating bond (two atoms in the bond, plus one extending from that bond, as in A - B - C and B - C - D in Figure 1.5) defines a plane. Thus, we can see from this definition that the torsion and dihedral angles are identical. Changing the conformation of a molecule does not make a new molecule, but can change its properties. The properly folded conformation of a protein, referred to as the native conformation, is its functional form, while the unfolded or denatured conformation is nonfunctional and often targeted for proteolysis by the cell. Thus, both the configuration and conformation of a molecule are important for its shape and function, but these represent distinct characteristics of the molecule and are not interchangeable terms. The conformations of polypeptides and polynucleic acids will be treated in greater detail in later sections. 1.2 MOLECULAR INTERACTIONS IN MACROMOLECULAR STRUCTURES The configurations of macromolecules in a cell are fixed by covalent bonding. The conformations, however, are highly variable and dependent on a number of factors. The sequence-dependent folding of macromolecules into secondary, tertiary, and quaternary structures depends on a number of specific interactions. This includes the interactions between atoms in the molecule and between the molecule and its environment. How these interactions affect the overall stability of a molecule and how they can be used to construct models of macromolecules are discussed in greater detail in Chapter 3. In this introductory chapter, we define some of the char­ acteristics of these interactions, so that we can have some understanding for how the various conformations of proteins and polynucleic acids are held together. 1.2.1 Weak Interactions The covalent bonds that hold the atoms of a molecule together are difficult to break, releasing large amounts of energy during their formation and concomitantly requiring large amounts of energy to break (Figure 1.6). For a stable macromole­ cule, they can be treated as invariant. The conformation of a macromolecule, how­ ever, is stabilized by weak interactions, with energies of formation that are at least one order of magnitude less than that of a covalent bond. The weak interactions de­ scribe how atoms or groups of atoms are attracted or repelled to minimize the en­ ergy of a conformation. These are, in general, distance-dependent interactions, with the energies being inversely proportional to the distance r or some power of the distance (r2, r3, etc.) separating the two interacting groups (Table 1.1). As the power of the inverse Section 1.2 Molecular Interactions in Macromolecular Structures 9 Figure 1.6 Energies of molecular interactions. The inter­ Si-O 1000 actions that define the structure of a molecule range from C-F the strong interactions of covalent bonds (200 to 800 C-H T kJ/mol) to the weak charge-charge (or ion-ion), dipole­ 500 _-'-_ C-OH Covalent dipole, dispersion, and hydrogen-bonding interactions bonds C-NH2 (0 to 60 kJ/mol). C-C C-I C-NO distance dependency increases, the interaction approaches zero more rapidly as r increases, and thus becomes a shorter range interaction. The interaction energy be­ tween two charges varies as lIr; this is a long-range interaction. At the other ex­ treme are the induced dipole-induced dipole (or dispersion) interactions. These interactions describe the natural tendency of atoms to attract, regardless of charge and polarity, because of the pol ariz ability of the electron clouds. Its dependence on 6 lIr defines this as a very short-range interaction, having a negligible interaction energy at about 1 nm or greater. Directly opposing this attraction, however, is steric repulsion, which does not allow two atoms to occupy the same space at the same time. This repulsion occurs at even shorter distances and is dependent on lIr12. Together, the attractive dispersion and repulsive exclusion interactions define an optimal distance separating any two neutral atoms at which the energy of inter­ action is a minimum. This optimal distance thus defines an effective radius (the van der Waals radius, or rvdw) for each type of atom. The potential energy functions for Table 1.1 Relationship of Noncovalent Interactions to the Distance Separating the Interacting Molecules, r Type of Interaction Distance Relationship Charge-charge l/r 2 Charge-dipole 1Ir 3 Dipole-dipole 1Ir 4 Charge-induced dipole 1Ir 6 Dispersion 1Ir 12 Repulsion l/rBiological Macromolecules Chapter 1 10 each interaction and their application to simulating the thermodynamic properties of macromolecules are treated in detail in Chapter 3. The energies associated with long-range interactions (charge-charge, charge­ dipole, and dipole-dipole) are dependent on the intervening medium. The interac­ tion between two charged atoms, for example, becomes shielded in a polar medium and is therefore weakened. The least polarizable medium is a vacuum, with a dielec­ 12 2 12 2 tric constant of KEO = 41T8.85 X 10- C J. m, where EO = 8.85 X 10- c J. m and K = 41T for a point charge. The polarizability of a medium is defined as its di­ electric constant D relative to that of a vacuum. The expressions for the energy of long-range interactions are all inversely related to the dielectric of the medium and are therefore weakened in a highly polarizable medium such as water. With the dielectric constant, we introduce the environment as a factor in stabi­ lizing the conformation of a macromolecule. How the environment affects the weak interactions is discussed in the next section. In the process, two additional interac­ tions (hydrogen bonds and hydrophobicity) are introduced that are important for the structure and properties of molecules. 1.3 THE ENVIRONMENT IN THE CELL The structures of macromolecules are strongly influenced by their surrounding en­ vironment. For biopolymers, the relevant environment is basically the solvent within the cell. Because the mass of a cell is typically more than 70% water, there is a ten­ dency to think of biological systems primarily as aqueous solutions. Indeed, a large majority of studies on the properties of biological macromolecules are measured with the molecule dissolved in dilute aqueous solutions. This, however, does not pre­ sent a complete picture of the conditions for molecules in a cell. First, a solution that is 70% water is in fact highly concentrated. In addition, the cell contains a very large surface of membranes, which presents a very different environment for macromole­ cules, particularly for proteins that are integral parts of the bilayer of the mem­ branes. The interface between interacting molecules also represents an important nonaqueous environment. For example, the recognition site of the TATA-binding protein involves an important aromatic interaction between a phenylalanyl residue of the protein and the nucleotide bases of the bound DNA. In cases where solvent molecules are observed at the molecular interfaces (for example, between the protein and its bound DNA), the water often helps to mediate interactions, but is often treated as part of the macromolecule rather than as part of the bulk solvent. In support of this, a well-defined network of water molecules has been observed to reside in the minor groove of all single-crystal structures of DNA duplex. Results from studies using nuclear magnetic resonance (NMR) spec­ troscopy indicate that the waters in this spine do not readily exchange with the bulk solvent and thus can be considered to be an integral part of the molecule. We start by briefly discussing the nature of the aqueous environment because it is the domi­ nant solvent system, but we must also discuss in some detail the nonaqueous envi­ ronments that are also relevant in the cell. Section 1.3 The Environment in the Cell 11 1.3.1 Water Structure Water plays a dominant role in defining the structures and functions of many mole­ cules in the cell. This is a highly polar environment that greatly affects the interactions within molecules (intramolecular interactions) and between molecules (intermolecular interactions). It is useful, therefore, to start with a detailed description of the :structure of water. A single H 0 molecule in liquid water is basically tetrahedral. The sp3 oxygen 2 atom is at the center of the tetrahedron, with the hydrogens forming two of the apices, and the two pairs of nonbonding electrons forming the other two apices (Figure 1.7). In the gas phase, the nonbonding orbitals are not identical (the spec­ troscopic properties of water are discussed in Chapter 9), but in the hydrogen­ bonded network they are. The oxygen is more electronegative than are the hydrogens (Table 1.2), leaving the electrons localized primarily around the oxygen. The 0 - H bond is therefore polarized and has a permanent dipole moment di­ rected from the hydrogen (the positive end) to the oxygen (the negative end). A di­ pole moment also develops with the positive end at the nucleus of the oxygen, pointing toward each of the nonbonding pairs of electrons. The magnitude of these dipoles becomes exaggerated in the presence of other charged molecules or other polar molecules. The magnitude of the dipole moment increases from 1.855 debye 30 (debye = 3.336 X 10- C/m) for an isolated water molecule to 2.6 debye in a clus­ ter of six or more molecules to 3 debye in ice. Water is therefore highly polarizable, as well as being polar. It has a very high dielectric constant relative to a vacuum (D = 78.5KEO)' The interaction between two water molecules is an interaction between polar compounds. This is dominated by the dipoles, which align the 0 - H bonds with the Figure 1.7 The structure of water. Each H 0 molecule has two hydrogens and two lone pairs of un­ 2 bonded electrons at each oxygen. In ice, the hydrogens act as hydrogen-bond donors to the lone pairs of the oxygens, which act as hydrogen-bond acceptors. This results in a hexagonal lattice of hydrogen­ bonded water molecules, with each H 0 molecule having four neighbors arranged in a tetrahedron. 2 Adapted from Mathews and van Holde (1996), Biochemistry, 2nd ed., 33. Benjamin-Cummings Co., Menlo Park, CA. 12 Biological Macromolecules Chapter 1 Table 1.2 Electronegativities of Elements Typically Found in Biological Molecules Element Electronegativity 0 3.5 Cl 3.0 N 3.0 S 2.5 C 2.5 P 2.1 H 2.1 2 Cu + 1.9 2 Fe + 1.8 Co2+ 1.8 Mg2+ 1.2 2 Ca + 1.0 Na+ 0.9 K+ 0.8 Higher values indicate a higher electron affinity. dipole moment of nonbonding electrons of the oxygen. These dipole-dipole interac­ tions bring the oxygen and hydrogen atoms closer than the sum of their van der Waal's radii, and thus are classified as weak bonds. This is the water-water hydrogen bond. In this case, the 0 - H donates the hydrogen to the bond and is the hydrogen­ bond donor. The oxygen, or more precisely the nonbonding electrons of the oxygen, acts as the hydrogen-bond acceptor. Water molecules therefore form a hydrogen­ bonded network, with each H 0 potentially donating hydrogen bonds to two neigh­ 2 bors and accepting hydrogen bonds from two neighbors. Other hydrogen-bond donors and acceptors that are important in biopolymers are listed in Table 1.3. The structure in which the H 0 molecules are exactly tetrahedral and uni­ 2 formly distributed into hexagonal arrays (Figure 1.7) is found only in the crystalline ice form of water. However, the hydrogens of the ice observed under normal condi­ tions (ice 1 at ODC and 1 atm pressure) remain disordered. They cannot be assigned to any particular oxygen at any given time, even though the oxygen atoms remain fixed. The hydrogens can be ordered precisely, but only at pressures greater than 20 kbars at temperatures less than ODC (Figure 1.8). This ice VllL therefore, forms only when work is performed against the inherent entropy in the hydrogens of the water molecules (see Chapter 2). Water can be induced at low temperatures and high pressures to adopt other forms or phases of ice that are unstable under normal conditions. The molecules in ice IX, for example, are arranged as pentagonal arrays. This arrangement is similar to many of the faces in the clathrate-like structures ob­ served around ions and alkyl carbons under standard conditions (Figure 1.9). The structure of liquid water is very similar to that of ice 1. This liquid form, which we will now simply refer to as water, is also a hydrogen-bonded network. The average stretching frequency of the 0 - H bond in water is more similar to that of Section 1.3 The Environment in the Cell 13 Table 1.3 Hydrogen-Bond Donors and Acceptors in Macromolecules Acceptor r(nm) Donor 0.29 H N-H···Do '" / 0.29 / . '" N/ N-H 0.31 '" / . '" / N-H 0.37 Ds '" / . '" () O-H 0.28 O '" / H G / O-H 0.28 0", / ice than to H 0 molecules that do not form hydrogen bonds (Figure 1.10). At the 2 air-water interface, the water molecules are well organized, much like that of ice, and form a highly cohesive network. A similar ordered structure is found at the in­ terface between water and the surface of molecules dissolved in water, which we de­ scribe in the next section. However, the structure of water is more dynamic than that of ice, with the pat­ tern of hydrogen bonds changing about every picosecond. The redistribution of pro­ tons results in a constant concentration of hydronium ions (a hydrate proton) and hydroxide ions in aqueous solutions, as defined by the equilibrium constant (Keq) (1.1) Biological Macromolecules Chapter 1 14 Pressure (kbar) 8 12 16 20 24 28 32 o 4 80 VII 40 -3 (I) 0 S (I) '" ..... -40 VI 2 ..... (I) " , -80 'ci • I 'II 1\ .1 I _\ I \ D " , -120 " , , ' , , : ilx -240 , , " , , ' Figure 1.8 Phase diagram for water. Liquid water freezes in different ice forms, depending on the tem­ perature and pressure. Under normal conditions, ice is a hexagonal network in which the protons of the hydrogen bonds are equally shared and cannot be assigned to a specific oxygen center (ice Ih). More compact forms (e.g., ice IX) or more ordered forms (e.g., ice VIII) are observed at low temperatures and high pressures. Adapted from H. Savage and A. Wlodawer (1986), Meth. Enzymol. 127; 162-183. Equation 1.1 is reduced to the standard equation for self-dissociation of water (1.2) with the concentration of H30+ represented by H+. Alternatively, this is given as (1.3) pKw = pH + pOH = 14 with panything = -loglOanything. Free protons do not exist in aqueous solution but are complexed with a local aggregate of water molecules. This is also true for the resulting hydroxide ion. The two ions are indeed distinct and have different properties, even in terms Figure 1_9 Clathrate structure of waters in the hydrated complex (nC H ghS+F-' 23 H 0. The solvent structure is 4 2 composed of regular hexagonal and pentagonal faces (one of eaeh is highlighted), similar to those found in ice structures. Adapted from G. L. Zubay, W. W. Parson, and D. E. Vance (1995), f'rinciples of BiachemLvtry, 14. Wm. C. Brown, Dubuque,IA.J Section 1.3 The Environment in the Cell 15 Figure 1.10 Vibrational frequency of 0 - H bond of H 0 in ice, in liquid water, and in 2 Liquid Water CCl . The vibration in CCl4 is very similar to 4 40 that of the bond in water vapor. Adapted from C. Tanford (1980), The Hydrophobic Ef­ J) fect: Formation of Micelles and Biological (.) 30 Membranes, 2nd ed., 36. John Wiley & Sons, cd .D New York. '"' 0 15 cd 20 cd '"' "0 :::;s 10 0 2800 3200 3600 4000 1 Frequency (cm- ) of the distribution of protons around each water molecule. A proton in HsOi sits at an average position between oxygens. In H 0 , the average distance between 2 3 oxygens is increased, leaving the shared proton distributed toward one or the other oxygen atom. The difference in the chemical properties of the two ionic forms of water may be responsible for the differences observed in how acids and bases affect biochemical reactions, particularly in the effects of deuterium or tri­ tium on the kinetics of enzyme-catalyzed reactions that require proton transfers. 1.3.2 The Interaction of Molecules with Water A molecule dissolved into water must interact with water. The polarizability of the aqueous medium affects the interactions between charged groups of atoms, polar but uncharged groups, and uncharged and nonpolar groups in macromolecules. These interactions are discussed in greater detail in Chapter 3. At this point, we will provide a general picture of how molecules interact with water, and how this affects the properties of the molecule as well as the properties of the solvent. When any molecule is placed in water, the solvent must form an envelope that is similar in many respects to the air-water interface. This is true whether the com­ pound is an ion or a hydrocarbon. Water molecules form a cage-like clathrate struc­ ture around ions (Figure 1.9). Compounds that can overcome the inherently low entropy of this envelope by interacting strongly with the water will be soluble. These hydrophilic compounds are water loving. Salts such as NaCI are highly soluble in water because they dissociate into two ions, Na+ and Cl-. The strong interaction be­ tween the charged ions and the polar water molecules is highly favorable, so that the net interaction is favorable, even with the unfavorable entropy contribution from the structured waters. 16 Biological Macromolecules Chapter 1 Hydrocarbons such as methane are neither charged nor polar, and thus are left with an inherently unfavorable cage of highly structured surrounding waters. This cage is ice-like, often with pentagonal faces similar to ice IX. These rigid ice-like cage structures are low in entropy and this is the primary reason that hydrocarbons are insoluble in water. These compounds are thus hydrophobic or water hating. The pentagonal arrays help to provide a curved surface around a hydrophobic atom, much like that seen at the air-water interface. The waters around hydrophilic atoms typically form arrays of six and seven water molecules. In contrast, hydrophobic molecules are highly soluble in organic solvents. Methane, for example, is highly soluble in chloroform. The favorable interactions between nonpolar molecules come from van der Waals attraction. Thus, polar and charged compounds are soluble in polar solvents such as water, and nonpolar com­ pounds are soluble in nonpolar organic solvents, such as chloroform. This is the basis for the general chemical principle that like dissolves like. Molecules that are both hydrophilic and hydrophobic are amphipathic. For ex­ ample, a phospholipid has a charged phosphoric acid head group that is soluble in water, and two long hydrocarbon tails that are soluble in organic solvents (Figure 1.11). In water, the different parts of amphipathic molecules sequester themselves into distinct environments. The hydrophilic head groups interact with water, while the hydrophobic tails extend and interact with themselves to form an oil drop-like hy­ drophobic environment. The form of the structures depends on the type of mole­ cules that are interacting and the physical properties of the system. In the case of phospholipids, the types of structures that form include micelles (formed by dilute dispersions), monolayers (at the air-water interface), and bilayers. A bilayer is par­ ticularly useful in biology as a membrane barrier to distinguish between, for exam­ ple, the interior and exterior environment of a cell or organelle. Proteins and nucleic acids are also amphipathic. Proteins consist of both polar and nonpolar amino acids, while nucleic acids are composed of hydrophobic bases and negatively charged phosphates. These biopolymers will fold into structures that resemble the structures of micelles. In general, molecules or residues of a macro­ molecule that are hydrophilic will prefer to interact with water, while hydrophobic molecules or residues will avoid water. This is the basic principle of the hydrophobic effect that directs the folding of macromolecules (such as proteins and nucleic acids) into compact structures in water. The basis for the hydrophobic effect and its role in stabilizing macromolecular structures is discussed in Chapter 3. 1.3.3 Nonaqueous Environment of Biological Molecules Several biologically important molecules exist in nonaqueous environments. These are predominantly molecules (primarily proteins) found in the phospholipid bilayers of cellular membranes. There are a number of significant differences between a cell membrane and the aqueous solution in a cell. The most obvious is that those parts of a molecule residing within the hydrocarbon tails of the membrane bilayer must be Section 1.3 The Environment in the Cell 17 a II 0-p - 0- CH - CH - NH, + 2 2 I a I CH -CH-CH 2 2 I I o 0 I I O=C C=O I I Monolayer HCH HCH "- "­ HCH HeH / / HCH HeH Micelle "- "­ HCH HeH / / HeH HCH Bilayer "- "­ vesicle HeH HCH / / HCH HCH "- "­ HCH HCH / / HCH HCH "- "­ HCH HCH / / HCH HCH "- "­ HCH HCH / / CH CH 3 3 Figure 1.11 Structures formed by amphipathic molecules in water. An amphipathic molecule. such as phosphatidyl choline, has a head group that is hydrophilic and long hydrophobic tails. In water, these compounds form mono lay­ ers at the water-air interface, globular micelles, or bilayer vesicles. Adapted from Mathews and van Holde (1996), Biochemistry, 2nd ed., 37. Benjamin-Cummings, Menlo Park, CA. hydrophobic. The structure of an integral membrane protein can be thought of as being inverted relative to the structure of a water-soluble protein, with the hydropho­ bic groups now exposed to the solvent, while the hydrophilic atoms form the internal­ ized core. An example of this inverted topology is an ion channel (Figure 1.12). The polar groups that line the internal surface of the channel mimic the polar water solvent, thus allowing charged ions to pass readily through an otherwise impenetrable bilayer. In addition to affecting the solubility of molecules, the organic nature of the hydrocarbon tails in a membrane bilayer makes them significantly less polar than water. Thus, the dielectric constant is approximately 40-fold lower than an aqueous solution. The effect is to enhance the magnitude of interactions dependent on D by a factor of about 40. One consequence of this dramatically lower dielectric constant is that the energy of singly charged ions in the lipid bilayer is significantly higher than that in aqueous solution. A measure of the energy of single charges in a partic­ ular medium is its self-energy Es. This can be thought of as the energy of a charge in the absence of its counterion and thus defined by an expression similar to that of a charge-charge interaction. (1.4) 18 Biological Macromolecules Chapter 1 Figure 1.12 (a) The crystal structures of the ion channel gramicidin and a calcium-binding ionophore A23187. Gramicidin is a left-handed antiparallel double helix in the crystal. In this structure, the central pore is filled with cesium and chloride ions. Adapted from B. A. Wallace and K. Ravikumar (1988), Science, 241; 182-187.) (b) The structure of A23187 binds a calcium by coordination to oxygen and nitrogen atoms. Adapted from Chaney (1976),.T. Antibiotics 29, 4.) Gramicidin (a) Calcium-binding ionophore A23187 (b) In this case, Es is dependent on the square of the single charge q2, and thus the self­ energy is always positive for single cations or anions. In this relationship, Rs is the Stokes' radius of the molecule (the effective molecular radius, see Section 5.2.1). The inverse relationship to the Stokes' radius Rs indicates that a charge that is distrib­ uted over a larger ion or molecule has a lower self-energy than an ideal point charge. 11le dependence of Es on 1ID means that the self-energy of an ion in water is 40 times lower than in a lipid bilayer. This translates into a probability that the ion will reside in the membrane is 1O-18 times that in water, thus making membranes highly efficient barriers against the passage of charged molecules. The movement of ions and other polar molecules through a cellular membrane requires the help of ion carriers, or ionophores, that form water-filled channels through the membrane or transport ions directly across the membranes (Figure 1.12). Membranes are also distinguished from an aqueous environment in that mem­ branes are essentially two-dimensional (2D) surfaces. With the exception of very small molecules such as the ionophores, molecules travel mostly in two dimensions in membranes. The concepts of concentration and diffusion-controlled kinetics must be defined in terms of this 2D surface, as opposed to a 3D volume. In solution, the 3 concentration of a molecule is given in units of moles/dm (moles/l = M). The con­ centration of a molecule in a membrane is defined as the number of molecules per given surface area (moles/dm2). For example, the concentration of molecules at the surface of a sphere will be diluted by a factor of 4 if the radius of that sphere is dou­ bled, while molecules within the volume of the sphere will be diluted by a factor of 8. Section 1.4 Symmetry Relationships of Molecules 13 The diffusional rates of molecules in aqueous solution and in membranes show these volume versus surface area relationships. Finally, it is not necessary for a molecule to be imbedded in a membrane to ex­ perience a nonaqueous environment. The interior of a globular protein consists pri­ marily of hydrophobic amino acids, and the polarizability of this environment is often compared to that of an organic solvent such as octanol. The consequence is that it is very difficult to bury a single charge in the interior of a large globular protein. This is reflected in a lower pKa for the side chains of the basic amino acids lysine and argi­ nine, or a higher pKa for the acidic amino acids aspartic acid and glutamic acid when buried in the interior of a protein. We can estimate the effect of the self-energy on the pKa of these amino acids in solution as opposed to being buried in the interior of a protein. We should reemphasize that these are estimates. A lysine with a pKa = 9.0 for the side chain would be protonated and positively charged in water. If we transfer this charged amino acid (with a Stokes' radius"" 0.6 nm) into a protein interior (D ::::: 3.5KEO), the difference in self-energy in the protein versus water IJ,.Es is about 40 kJ/moi. We can treat this energy as a perturbation to the dissociation constant by (1.5) and predict that the pKa 1 for a lysine buried in the hydrophobic core of a globu­ lar protein and therefore would be uncharged unless it is paired with a counterion such as an aspartic acid residue. 1.4 SYMMETRY RELATIONSHIPS OF MOLECULES Biological systems tend to have symmetry, from the shape of an organism to the structure of the molecules in that organism. This is true despite the fact that the monomer building blocks (amino acids, for example) are always asymmetric. Yet, these often combine to form elegantly symmetric structures. In this section, we de­ scribe the symmetry relationships of biological macromolecules, both conceptually and mathematically. This mathematical formalism provides a means to precisely overlay or map two symmetry-related objects on top of each other, or to construct a set of symmetry-related objects from a starting model. This is useful in that it simpli­ fies many problems in physical biochemistry, including structure prediction, struc­ ture determination using techniques such as X-ray diffraction and electron diffraction, and image reconstruction that improves the results obtained from elec­ tron microscopy or atomic microscopies. Symmetry is the correspondence in composition, shape, and relative position of parts that are on opposite sides of a dividing line or median plane or that are dis­ tributed about a center or axis (Figure 1.13). It is obvious that two or more objects are required in order to have a symmetric relationship. The unique object in a sym­ metry-related group is the mqtif. A motif m is repeated by applying a symmetry ele­ ment or symmetry operator 0 to give a related motif m'. (1.6) Oem) = m'

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