For a typical protein, the global stability at room temperature is of the order 10 kcal/mol ∼20 kBT for a ∼100 residue domain. Thus proteins are stabilized with a free energy of a fraction of kBT per amino acid. Nonetheless, proteins have well-defined structures that are stabilized by cooperative interactions between their many small parts. Thus there is essentially no “partial melting” of the protein; it stays folded and specifically functional to do whatever job it is supposed to do. When a large enough fluctuation does indeed melt the protein, the structure can (mostly) reassemble.

Themiracle of protein structure and function in the noisy Brownian environment of ongoing kBT perturbations may be envisioned as a car engine exposed to perturbations that could easily tear any part of it at any time. Imagine that a car would melt at maybe 70°C, but would stay together and remain functional nearly all the time. However, should it disassemble, it would reassemble by itself and resume functioning. Proteins may be regarded as “intelligent” and robust nano-technology made of programmable matter.

Proteins can fold into their native state in a remarkably short time; one infers from this that the enormous phase space available to the polypeptide chain must be sampled not randomly, but following one or more “folding pathways”. Thus the amino acid sequence contains not only the plan for the final structure, but it also specifies how to get there (see Fig. 5.1).

This book covers some subjects that we find inspiring when teaching physics students about biology. The book presents a selection of topics centered around the physics/biology/chemistry of genes. The focus is on topics that have inspired mathematical modeling approaches. The presentation is rather condensed, and demands some familiarity with statistical physics from the reader. However, we attempted to make the book complete in the sense that it explains all presented models and equations in sufficient detail to be self-contained. We imagine it as a textbook for the third or fourth years of a physics undergraduate course.

Throughout the book, in particular in the introductions to the chapters, we have expressed basic biology ideas in a very simplified form. These statements are meant for the physics student who is approaching the biological subject for the first time. Biology textbooks are necessarily more descriptive than physics books. Our simplified statements are meant to reduce this difference in style between the two disciplines. As a consequence, the expert may well find some statements objectionable from the point of view of accuracy and completeness. We hope, however, that none is misleading. One should think of these parts as first-order approximations to the more complicated and complete descriptions that molecular biology textbooks offer. On the other hand, the physical reasoning that follows the simplified presentation of the biological system is detailed and complete.

Evolution and evolvability

Evolution, and the ability to evolve, is basic to life. In fact, it separates life from non-life. The process of evolution has brought us from inorganic material in ∼3.5 billion years, our bacterial ancestors in ∼2 billion years, from primitive seadwelling chordates in ∼500 million years and from common ancestors to mice in only ∼100 million years. Evolution has inspired our way of looking at life processes throughout modern biology, including the idea that evolvability is an evolving property in itself.

Most mutations are either without any consequences, i.e. neutral, or deleterious. Thus evolution is a costly process where most attempts are futile. That it works anyway reflects the capacity of life to copy itself so abundantly that it can sustain the costly evolutionary attempts.

Evolution is the necessary consequence of:

• heredity (memory);

• variability (mutations);

• each generation providing more individuals than can survive (surplus).

The last of these points implies selection, and thus the principle of “survival of the fittest”. In fact, the surplus provided by growth of successful organisms is so powerful that it provides a direct connection between the scale of the molecule and the worldwide ecological system: a successful bacterium with a superior protein for universal food consumption may in principle fill all available space on Earth within a few days. In fact, bacteria on Earth number about 5 × 1030 (Whitman et al., 1998), a biomass that dominates all other animal groups together. The bacteria have already won! The dynamics of this interconnected microbiological ecosystem thus challenges any model built on a separation of phenomena at small and larger length scales.

Living cells consist of a wide variety of molecular machines that perform work and localize this work to the proper place at the proper time. The basic design idea of these nano-machines is based on a one-dimensional backbone, a polymer. That is, these nano-machines are not made of cogwheels and other rigid assemblies of covalently interlocked atoms, but rather are based on soft materials in the form of polymers – i.e. one-dimensional strings. In fact most of the macromolecules in life are polymers. Along a polymer there is strong covalent bonding, whereas possible bonds perpendicular to the polymer backbone are much weaker. Thereby, the covalent backbone serves as a scaffold for weaker specific bonds. This opens up the possibility (1) to self-assemble into a specific functional three-dimensional structure, (2) to allow the machine parts to interact while maintaining their identity, and (3) to allow large deformations. All three properties are necessary ingredients for parts of a machine on the nano-scale. In this chapter we review the general properties of polymers, and thus hope to familiarize the reader with this basic design idea of macromolecules.

Almost everything around us in our daily life is made of polymers. But despite the variety, all the basic properties can be discussed in terms of a few ideas. Some of these properties are astounding: consider a metal wire and a rubber band. The metal wire can be stretched about 2% before it breaks; its elasticity comes from small displacements of the atoms around a quadratic energy minimum.

DNA

The molecular building blocks of life are DNA, RNA and proteins. DNA stores the information of the protein structure, RNA participates in the assembly of the proteins, and the proteins are the final devices that perform the tasks. Two other prominent classes of molecule are lipids (they form membranes and, thus, compartments), and sugars (they are the product of the photosynthesis; ultimately life depends on the energy from the Sun). Sub-units from these building blocks show up in different contexts. For example ATP is the ubiquitous energy storing molecule, but adenine (the A in ATP) is also one of the DNA bases; similarly, the sugar ribose is also a component of the DNA backbone.

Before entering into the details of the basic building units, let us list their respective sizes. We do this with respect to their information content. DNA is used as the main information storage molecule. The information resides in the sequence of four bases: A, T, C and G. Three subsequent bases form a codon, which codes for one of the 20 amino acids. The genetic code is shown in Table 3.1. The weight of a codon storage unit is larger for DNA (double stranded and 330 Da per nucleotide, 1 Da = 1 g/mol) than for RNA (single stranded) and proteins.

1. DNA: 2000 Da per codon,

2. RNA: 1000 Da per codon,

3. proteins: 110 Da per amino acid.

Thus whereas a 300 amino acid protein weighs 33 kg/mol, the stored information at the DNA level has a mass of 600 kg/mol.

Life is self-reproducing, persistent (we are ∼ 4 × 109 years old), complex (of the order of 1000 different molecules make up even the simplest cell), “more” than the sum of its parts (arbitrarily dividing an organism kills it), it harvests energy and it evolves. Essential processes for life take place from the scale of a single water molecule to balancing the atmosphere of the planet. In this book we will discuss the modeling and physics associated, in particular, to the molecules of life and how together they form something that can work as a living cell. First we briefly review some basic concepts of living systems, with emphasis on what makes biological systems so different from the systems that one normally studies in physics.

Conceptually, molecular biology has provided us with a few fundamental/universal mechanisms that apply over and over. Some concepts, like evolution, do not have counterparts in physics. Others, like the role of stochastic processes, are, on the contrary, quite familiar to a physicist.

1. (1) Biology is the result of a historical process. This means that it is not possible to “explain” a biological system by applying a few fundamental laws in the same way that is done in physics. A hydrogen atom could not be different from what it is, based on what we know of the laws of nature, but an E. coli cell could. In evolution, it is much easier to modify existing mechanisms than to invent new ones. Thus on evolutionary timescales nearly everything comes about by cut and paste of modules that are already working. We will end the book with a chapter dedicated to evolutionary concepts and models.

2. […]

This book was initiated as lecture notes to a course in biological physics at Copenhagen University in 1998–1999. In this connection, Chapters 1–5 were developed as a collaboration between Kim Sneppen and Giovanni Zocchi. Later chapters were developed by Kim Sneppen in connection to courses taught at the Norwegian University of Science and Technology at Trondheim (2001) and at Nordita and the Niels Bohr Institute in 2002 and 2003.

A book like this very much relies on feedback from students and collaborators. Particular thanks go to Jacob Bock Axelsen, Audun Bakk, Tom Kristian Bardøl, Jesper Borg, Petter Holme, Alexandru Nicolaeu, Martin Rosvall, Karin Stibius, Guido Tiana and Ala Trusina. In addition, much of the content of the book is the result of collaborations that have been published previously in scientific journals. Thus we would very much like to thank:

• Jesper Borg, Mogens Høgh Jensen and Guido Tiana for collaborations on polymer collapse modeling;

• Terry Hwa, E. Marinari and Lee-han Tang for collaborations on DNA melting;

• Audun Bakk, Jacob Bock, Poul Dommersness, Alex Hansen and Mogens Høgh Jensen for collaborations on protein folding models and models of discrete ratchets;

• Deborah Kuchnir Fygenson and Albert Libchaber for collaborations on nucleation of microtubules and inspiration;

• Erik Aurell, Kristoffer Bæk, Stanley Brown, Harwey Eisen and Sine Svenningsen for collaborations on λ-phage modeling and experiments;

• Ian Dodd, Barry Egan and Keith Shearwin for collaborations on modeling the 186 phage;

• […]

Biological molecular systems work inside living cells. As a cell prototype we consider the prokaryote Escherichia coli. This may be viewed as a small bag of DNA, RNA and proteins, surrounded by a membrane. The bag has a volume of about 1 μm3. This volume varies as the cell grows and divides, and also varies in response to external conditions such as osmotic pressure. The interior of the cell is a very crowded environment, with about 30% to 40% of its weight in proteins and other macromolecules, and only about 60% as water. Further, the water contains a number of salts, in particular K+, Cl- and Mg2+, each of which influences the stability of different molecular complexes.

The dry weight of E. coli consists of 3% DNA, 15% RNA and 80% proteins. The genome of E. coli is a singleDNAmolecule with 4.6 × 106 base pairs (total length of about 1.5 mm). It codes for 4226 different proteins and a number of RNA molecules. However, the information content of the genome is larger than that corresponding to the structure of the coded macromolecules. Important information is hidden and resides in the regulation mechanisms that appear when these proteins interact with the DNA and with each other. Some proteins, called transcription factors, regulate the production of other proteins by turning on or off their genes. Figure 7.1 shows two ways by which a transcription factor can regulate the transcription of a gene. The figure also shows a specific example of a regulatory protein bound to the DNA: the CAP protein.

Molecular motors: energy from ATP

Proteins can be grouped into a few broad categories with respect to their function. Some are regulatory, some are enzymes, some are structural, and some proteins do mechanical work. It is this latter group that we now discuss. Molecular motors include kinesin, myosin, dynein, the motors connected to DNA replication, to gene transcription and to translation. The motors are mostly driven by ATP hydrolysis: ATP → ADP + P, a process with ΔG ≈ 13 kcal/mol for typical conditions in the cell. Exactly how the free energy difference from ATP hydrolysis is converted into directed motion and mechanical work is a most interesting question, which is not resolved. In many cases the conformational changes of the protein are known in considerable detail from structural studies. The sequence of events associating conformational changes and substrate binding and release is also known. Nonetheless, the actual physical mechanism by which the motor works is not obvious. Thermal noise and diffusion certainly play a role, making this “soft” machine qualitatively different from a macroscopic motor. In the next section we elaborate on these ideas through some models.

The most studied motors include myosin and kinesin, which move along the polymers that define the cytoskeleton. Kinesin walks on microtubules (Fig. 6.1), whereas myosin walks on polymerized actin. Microtubules and actin fibers are long (μm) polymers where the monomer units are proteins. Microtubules are very stiff; actin fibers more flexible. Kinesin motors work independently of each other, and are associated with the transport of material (vesicles) inside the eukaryotic cell.

Cells are controlled by the action of molecules upon molecules. Receptor proteins in the outer cell membrane sense the environment and may subsequently induce changes in the states of specific proteins inside the cell. These proteins then again interact and convey the signal further to other proteins, and so forth, until some appropriate action is taken. The states of a protein may, for example, be methylation status, phosphorylation or allosteric conformation as well as sub-cellular localization. The final action may be transcription regulation, thereby making more of some kinds of proteins, it may be chemical, or it may be dynamical. A chemical response would be to change the free concentration of a particular protein by binding it to other proteins. A dynamical response could be the activation of some motor, as in the chemotaxis of E. coli.

The presently known regulatory network of yeast is shown in Fig. 8.1. The action of proteins in this network is to control the production of other proteins. The control is done through genetic regulation discussed previously, through control of mRNA degradation, or possibly through the active degradation of the proteins.

Regulatory genetic networks are essential for epigenetics and thus for multicellular life, but are not essential for life. In fact, there exist prokaryotes with nearly no genetic regulation. Figure 8.2 shows the number of regulators as a function of genome size for a number of prokaryotic organisms. One notices that those with a very small genome hardly use transcriptional regulation.

With some notable exceptions such as DNA, most filaments in the cell are linked together as part of a network, which may extend throughout the interior of the cell, or be associated with a membrane as an effectively two-dimensional structure such as shown in Fig. 3.1. This chapter concentrates on the properties of planar networks, particularly the temperature- and stress-dependence of their geometry and elasticity. To allow sufficient time to develop the theoretical framework of elasticity, we focus in this chapter only on networks having uniform connectivity.

Soft networks in the cell

Two-dimensional networks arise in a variety of situations in the cell; they may be attached to its plasma or nuclear membrane, or be wrapped around a cell as its wall. Containing neither a nucleus nor other cytoskeletal components such as microtubules, for example, the human red blood cell possesses only a membrane-associated cytoskeleton. Composed of tetramers of the protein spectrin, the erythrocyte cytoskeleton is highly convoluted in vivo (see Fig. 2.21(b)), but can be stretched by about a factor of seven in area to reveal its relatively uniform four-to six-fold connectivity, as shown in Fig. 3.1(a) (Byers and Branton, 1985; Liu et al., 1987; Takeuchi et al., 1998). Roughly midway along their 200 nm contour length, the spectrin tetramers are attached to the plasma membrane by the protein ankyrin (using another protein called band 3 as an intermediary).