Molecular and Cellular Signaling - Martin Beckerman

82 4. Macromolecular Forces

FIGURE 4.3. Bond bending: Plotted are the potential energies associated with rotations of the angle made by three atoms covalently bonded to one another. In a manner analogous to bond stretching, there is an equilibrium bond angle where the potential energy is at a minimum. As the rotation angles are either widened out or squeezed in, the potential energy rapidly increases. The bending spring constant Kq = 40 kcal/(mol · rad2), and equilibrium angle qeq = 120 deg.

FIGURE 4.4. Bond torsions: Plotted are the periodic contribution to the potential energy from torsion (dihedrals) rotations of covalently bonded sets of four atoms. Calculations were done using a Kj with four paths so that 3.625 kcal/mol is the constant in front of the expression. Parameters values used were n = 2 and d = 180 deg.

involving the twelfth power of the radius is a repulsive one. As can be seen in the figure there are two distances of interest. One of these is the equilibrium radius r0, the location where the potential has its minimum and where the force—the derivative of the potential—is zero. The other key dis-

4.12 Moleculer Dynamics in the Study of System Evolution

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FIGURE 4.5. Van der Waals interactions: Plotted are the van der Waals potential energies in kcal/mol using a well depth e = 0.19 kcal/mol and a radius r0 = 3.59 Å, corresponding to nitrogen and oxygen contact radii of 1.7 + 1.5 = 3.2 Å.

tance is the radius at which the repulsion exactly cancels out the attraction so that the net potential is zero. This repulsion is generated by the interpenetrating electron clouds. This contact distance is the van der Waals radius. As can be seen in the plot, any further interpenetration is strongly resisted by the rapidly increasing repulsive force arising from the Pauli exclusion.

Ï

È

r

12

r

6

˘

q q

j

¸

Unonbonded = ÂÌe ÍÊ

0

ˆ

- 2Ê

0

ˆ

˙

+

i

˝

(4.3)

i π j Ó

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Ë r

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Hydrogen-bonding contributions

to

the

total

potential are

treated in

an approximate way in the MM formalism. They are modeled using Lennard–Jones and coulombic like terms suitably modified to match the characteristics of the hydrogen bonds.

4.12Moleculer Dynamics in the Study of System Evolution

The evolution of macromolecular systems over time can be studied using the method of moleculer dynamics (MD). In molecular dynamics, the classical (Newtonian) equations of motion are solved through numerical

84 4. Macromolecular Forces

integration. The forces F are the spatial derivatives of the potential energies U presented in Eqs. (4.2) and (4.3). These quantities are then converted to accelerations using Newton’s second law, F = ma, where m is mass and a is acceleration. Once this is done the equations of motion are solved to give a picture of how the system evolves over time. Numerical methods are used to convert the equations of motion into a form suitable for numerical integration on a computer. A number of computer programs are in use that enable the researcher to carry out MD simulations.

The last entry in the list of methods is quantum mechanics (QM). The covalent bonds are only handled in an approximate way in the MM formalism. In a quantum mechanical approach, the empirical treatment of the covalent bonds through the use of a Hooke’s law is replaced by an exact quantum mechanical treatment. The quantum mechanical method is exact and rigorous but is difficult to use in practice due to computational limitations. As computer technologies have advanced, quantum mechanical studies have become more numerous. An often-used approach is to combine the MM and QM formalisms, using one or the other of the two methods for different aspects of the system under study.

4.13Importance of Water Molecules in Cellular Function

Water molecules are essential components of protein, DNA, and RNA function. Water plays a central role in protein structure and function. In the absence of water, a protein would not be able to fold into its native state, nor would it be able to catalyze reactions. Water is an essential component of many protein-protein, protein-DNA and protein-RNA interfaces, and without water proteins would not be able to recognize their substrates.

Water surrounds a protein, fills in pockets and grooves on the surface, and occupies voids in the interior.

When a protein is placed into a water environment it alters the network of hydrogen bonds. The water molecules in the vicinity of the protein surface reorient themselves so that positive and negative regions of change are in opposition. The rotations of the water molecules disrupt the hydrogen bonds between adjacent water molecules and thus alter the network. The water molecules in the immediate vicinity of the protein surface that have reoriented themselves are referred to as the first hydration layer. The reorientation effect propagates outward from the protein and the next layer of disrupted hydrogen bonded water molecules is designated as the second hydration layer.

The same phenomena occur for DNA and RNA. Water molecules in the vicinity of the DNA and RNA form hydrogen bonds to the molecules. These bonds are not passive entities but instead contribute to the conformational stabilization and function of the macromolecules. Hydrogen bonding net-

References and Further Reading

85

works between water and DNA is essential for DNA stability. DNA denatures as it dehydrates. Hydrogen bonds between water and ssRNA are even more numerous than in the case of dsDNA because of the single strand character of the RNA leaves bases unpaired and there are additional ribose oxygen atoms available for bonding.

4.14 Essential Nature of Protein Dynamics

Macromolecules such as proteins are dynamic entities: their internal motions are essential. They are in continual thermal motion and through these motions are continually exploring different conformations. When especially stable states are encountered, the dwell time, the period of time that the protein remains in such states, increases; when highly unstable states are populated the dwell time decreases. This continual exploration of available conformations is central to binding and catalysis.

At the heart of the role of water in the activities of all macromolecules is its ability to promote rapid conformational changes. Water is a common element in the active site of enzymes, and is a key mediator of the catalytic activity of many, if not most, enzymes. When dehydrated these enzymes lose their catalytic abilities. Yet another place where water seems to be crucial is in regulation. Proteins such as hemoglobin that use allosteric mechanisms (this term will be defined and its properties explored in the next chapter) for regulation operate in a hydrated fashion, and loss of these water molecules impairs the regulatory functioning of the protein. Water, the hydrogen bonds that are continually being made and broken, and the underlying thermal agitation collectively serve as a “lubricant” that promotes conformation changes essential to the performance of protein, DNA, and RNA functions.

General Reference

Brandon C, and Tooze J [1999]. Introduction to Protein Structure, 2nd edition. New York: Garland Science Publishing.

References and Further Reading

Physical and Electrostatic Properties of Amino Acids and Proteins

Bolen DW, and Baskakov IV [2001]. The osmophobic effect: Natural selection of a thermodynamic force in protein folding. J. Mol. Biol., 310: 955–963.

Dill KA [1990]. Dominant forces in protein folding. Biochem., 29: 7133–7155. Myers JK, and Pace CN [1996]. Hydrogen bonding stabilizes globular proteins.

Biophys. J., 71: 2033–2039.

Pace CN, et al. [1996]. Forces contributing to the conformational stability of proteins. FASEB J., 10: 75–83.

86 4. Macromolecular Forces

Sheinerman FB, and Honig B [2002]. On the role of electrostatic interactions in the design of protein-protein interfaces. J. Mol. Biol., 318: 161–177.

Tsai J, et al. [1999]. The packing density in proteins: Standard radii and volumes. J. Mol. Biol., 290: 253–266.

Complementarity and Interfaces

Glaser F, et al. [2001]. Residue frequencies and pairing preferences at proteinprotein interfaces. Proteins, 43: 89–102.

Lo Conte L, Chothia C, and Janin J [1999]. The atomic structure of protein-protein recognition sites. J. Mol. Biol., 285: 2177–2198.

Jones S, and Thornton JM [1996]. Principles of protein-protein interactions. Proc. Natl. Acad. Sci. USA, 93: 13–20.

Jones S, et al. [1999]. Protein-DNA interactions: A structural analysis. J. Mol. Biol., 287: 877–896.

Nadassy K, Wodak SJ, and Janin J [1999]. Structural features of protein-nucleic acid recognition sites. Biochem., 38: 1999–2017.

Norel R, et al. [1999]. Examination of shape complementarity in docking of unbound proteins. Proteins, 36: 307–317.

Sheinerman FB, Norel R, and Honig B [2000]. Electrostatic aspects of proteinprotein interactions. Curr. Opin. Struct. Biol., 10: 153–159.

Hot Spots

Bogan AA, and Thorn KS [1998]. Anatomy of hot spots in protein interfaces. J. Mol. Biol., 280: 1–9.

Hu ZJ, et al. [2000]. Conservation of polar residues as hot spots at protein interfaces. Proteins, 39: 331–342.

Theoretical Methods: Computer Modeling and Simulation

Cornell WD, et al. [1995]. A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J. Am. Chem. Soc., 117: 5179– 5197.

Elcock AH, Sept D, and McCammon JA [2001]. Computer simulation of proteinprotein interactions. J. Phys. Chem. B, 105: 1504–1518.

Honig B, and Nicholls A [1995]. Classical electrostatics in biology and chemistry. Science, 268: 1144–1149.

Kollman PA, et al. [2000]. Calculating structures and free energies of complex molecules: Combining molecular mechanics and continuum models. Acc. Chem. Res., 33: 889–897.

Problems

4.1Atomic motions. Atoms in a protein are constantly in thermal motion. Assuming an average energy kT of 0.6 kcal/mol for each atom, how fast is a hydrogen atom moving? How fast is a carbon atom moving? How long will it take for each of these atoms to move 1 Å, roughly one bond length?

Problems 87

4.2Numerical integration. Numerical techniques, known as finite difference methods, are used to convert the equations of motion into a form suitable for numerical integration on a computer. The basic idea is to take the position and momentum of each particle at a given time and compute how each changes over a small interval of time, the time step Dt, by calculating the accelerations from the forces, and these from the potentials of the form given in the chapter. In other words, for each particle i in the system

ai = 1 Fi = 1 d Ui . mi mi dri

A number of computer programs such as CHARMM, AMBER and GROMOS, are in use that enable a user to carry out MM simulations. A number of time-stepping algorithms are employed in determining the future positions from the past positions and forces. These algorithms are based on expansions such as

r(t + Dt) = r(t) + v(t)Dt + (1 2)a(t)(Dt)2 .

One of the most widely used stepping forms, known as the Verlet algorithm, is

r(t + Dt) = 2r(t) - r(t - Dt) + a(t)(Dt)2 .

Note that the velocities do not appear in this expression. The positions at time t + Dt are computed from the positions at the present (t + Dt) and previous (t) times and from the accelerations at the previous (t) time. Some algorithms use the velocities explicitly in the computations. One of these is the velocity Verlet algorithm. Its form is

v(t + Dt) = v(t) + (1 2)[a(t) + a(t + Dt)]Dt.

By making use of the appropriate expansions, and combining terms, derive both of these Verlet algorithms. What might be an appropriate time step size? (Hint: Think about the results from Problem 4.1.)

5

Protein Folding and Binding

The world contains a myriad of biological systems. All exhibit a considerable degree of order. They are organized in a hierarchical manner, and order is present at all levels of the hierarchy. From hydrogen, carbon, nitrogen, and oxygen, simple atomic groups are formed such as methyl (CH3) and hydroxyl (OH). These groups are then used to form the basic building blocks of cells—sugars, fatty acids, nucleotides, and amino acids. Simple sugars (monosaccharides) are organized into short chains (oligosaccharides) or longer ones (polysaccharides). Fatty acids form complexes such as triglycerides and phospholipids. Nucleotides are used to make RNA and DNA, and the amino acids give rise to polypeptides, or proteins.

Biological order does not arise out of some mysterious “vital force,” but rather is a consequence of the laws of thermodynamics and the character of the forces in our universe, their strength and their dependence on distance. At first glance the emergence of highly organized biological entities seems at odds with the second law of thermodynamics. This law establishes a thermodynamic arrow of time—the total disorder in the universe increases as the universe ages, until a terminal stage of disorder is reached in which the universe suffers a heat or entropy death. Yet, this first impression is wrong: Order comes about not in spite of the laws of physics but rather because of them.

Biological systems are open, continually exchanging matter and energy with their surroundings. They generate order by taking in energy and releasing heat to their surroundings. They absorb radiant energy from sunlight and from geothermal sources, and they take in foodstuffs that store energy in high energy chemical bonds. According to the second law of thermodynamics the amount of entropy, or disorder, in a cell and its surroundings must increase during any process. Thus, the production of order within a cell is accompanied by the creation of a greater amount of disorder outside a cell. This is accomplished through the release of heat from the cell at the same time that the order is produced. Biological entities are not only highly ordered, but actively generate these states in order to survive and propagate.

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90 5. Protein Folding and Binding

One of the central order-creating processes in a cell is the folding of a protein into its biologically active three-dimensional form. During folding, nascent proteins change their shape from a rather stretched out configuration to a highly compact form. They develop their secondary structure— alpha helices and beta sheets—and higher order structures with well-defined signaling roles such as binding motifs and functional domains. The folding process is the main focus of this chapter. Starting with a brief review of the thermodynamic conditions for order to emerge, the spontaneous folding of proteins will be examined. Large and complex proteins, especially those involved in signaling, often require the assistance of a class of molecules called molecular chaperones to fold and to maintain their correct form in the cell. Chaperone-assisted folding will be explored next. That topic will be followed by the third and final topic in this chapter, the relationship between the thermodynamic properties of the low-lying stable states of the folded proteins and their binding and signaling activities.

5.1The First Law of Thermodynamics: Energy Is Conserved

The first law of thermodynamics is expressed in terms of three factors: the internal energy of a system, the work done by that system, and the heat absorbed by a system from its surroundings. The internal energy of a system is the sum of the kinetic and potential energies of its constituents. Work is done whenever a force is applied to an object to produce a displacement. Typical examples of systems doing mechanical work are pistons, levers, and pulleys. The most commonly encountered form of work in chemical systems is pressure-volume work. In these systems, pressure, or force per unit area, is usually held constant and there is a change in the volume occupied by the system doing the work. If two systems are in thermal contact with one another, energy will flow from the hotter (higher temperature) system to the colder (lower temperature) system. “Heat” is the designation given to the flow, or transfer, of (thermal) energy from one system to another due to a temperature gradient.

In a chemical reaction that takes place in a cell, or in any other system, the internal energy E is lowered by the amount of energy used to do useful work and increased by the amount of heat absorbed in the process. In more detail, as a system evolves from state a to state b, its internal energy will change by an amount dE = Eb - Ea. If the amount of work done by the system on its surroundings is written as W, and the amount of heat absorbed by the system from its surroundings is designated as Q, then the first law of thermodynamics states that

dE = Q - W.

(5.1)

Energy will be gained by a system whenever energy flows into the system due to temperature gradients and whenever the surroundings do work on

5.2 Heat Flows from a Hotter to a Cooler Body

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the system. Energy will be lost from a system whenever the system does work on its surroundings and whenever energy is lost to the surroundings due to thermal gradients. If we consider pressure-volume work then we may write this as

dE = Q - PdV ,

(5.2)

where P is the pressure and dV = Vb - Va is the change in volume produced by the application of the (constant) pressure.

5.2 Heat Flows from a Hotter to a Cooler Body

Heat is associated with random molecular motion. If a hotter body is in contact with a cooler one in a way that allows matter and energy to flow from one to the other, temperature differences will be reduced and eventually vanish. The reason for this is a statistical one. There are many more ways that the contributions of energies of the randomly moving molecules can add up to a particular internal energy when the temperatures have equilibrated than when there are large temperature differences between parts of the whole. If, through random motions or perturbations of the individual molecules, one portion of the system gains an appreciable amount of thermal energy at the expense of the rest it will not retain it for long. Instead, the system will relax back to a thermally equilibrated distribution. The thermally equilibrated distribution thus has the property that it is the stable distribution. It is also the maximally disordered distribution, in contrast to highly ordered situations where each piece of a system is at some specific value of the temperature.

These everyday observations are depicted in Figure 5.1. Part (a) of the

figure illustrates mass equilibration and part (b) the similar process of thermal equilibration. In the case of mass equilibration a system starts out with most of its mass concentrated in one of two interconnected compartments. The particles are free to diffuse, and over time the masses become far more evenly distributed between the two compartments. The statistical character of the process is easy to see. Situations (states) where all or mostly all of the mass is concentrated in one compartment are rare. In contrast, there are many ways of distributing half the mass in one compartment and half in the other, and so under the influence of random movements of mass these partitions will occur most, all the time. Situations where all the particles are in one compartment will rarely occur, and when they do the system will not remain so for long (these states are not stable ones).

The same reasoning applies to thermal equilibration. The rare velocity distribution, where all the fast particles are in one compartment and the slow ones in the other, is replaced over time by the usual one where both compartments containing similar mixes of fast and slow movers. Again, there are many ways of achieving this kind of distribution and few for the

92 5. Protein Folding and Binding

FIGURE 5.1. Mass and thermal equilibration: (a) Mass equilibration—In the left panel, most of the mass is concentrated in the left compartment. The particles are free to diffuse to and fro, and through the opening into the adjoining compartment. Over time, the system will mass equilibrate. In the right panel, roughly half of the particles are in each compartment. (b) Thermal equilibration—Arrows denote velocity; particles with longer arrows are moving with a greater velocity than those with shorter arrows. In the left panel, all of the fast particles are in the left compartment, and all the slow particles are in the right compartment. As a result, the temperature in the left compartment is higher than that in the right compartment. Again, there is an opening, and the particles can freely move about. Over time, the system equilibrates so that the temperature in each chamber becomes the same. In the right panel, each compartment contains a similar mix of slow and fast moving particles.

other. In the case of thermal equilibration, the temperatures in the two compartments initially differ. The compartment with the fast movers is at a higher temperature than that containing the slow movers, but over time the temperatures in the two compartments become the same.

5.3Direction of Heat Flow: Second Law of Thermodynamics

The second law of thermodynamics formalizes the observation that heat flows from a hotter system to a cooler one, and not vice versa. The quantity called “entropy” gives a measure of the number of ways the molecular constituents can arrange themselves (i.e., it counts the number of microstates) to achieve a particular value of the macroscopic internal energy. The entropy increases as the particles approach the distribution that can be achieved in the largest number of ways, and entropy is maximal when the system equilibrates over time. The maximization process is interpreted as a flow of heat.